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10404-h-m-hypebolic-la-gi [2018/11/07 17:08] (current)
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 +<​HTML><​br><​div id="​mw-content-text"​ lang="​vi"​ dir="​ltr"><​div class="​mw-parser-output"><​div class="​thumb tright"><​div class="​thumbinner"​ style="​width:​298px;"><​img alt=""​ src="​http://​upload.wikimedia.org/​wikipedia/​commons/​thumb/​b/​bc/​Hyperbolic_functions-2.svg/​296px-Hyperbolic_functions-2.svg.png"​ width="​296"​ height="​259"​ class="​thumbimage"​ srcset="//​upload.wikimedia.org/​wikipedia/​commons/​thumb/​b/​bc/​Hyperbolic_functions-2.svg/​444px-Hyperbolic_functions-2.svg.png 1.5x, //​upload.wikimedia.org/​wikipedia/​commons/​thumb/​b/​bc/​Hyperbolic_functions-2.svg/​592px-Hyperbolic_functions-2.svg.png 2x" data-file-width="​500"​ data-file-height="​437"/> ​ </​div></​div>​
 +<​p>​Trong toán học, <​b>​hàm hyperbolic</​b>​ có những tính chất tương tự như các hàm lượng giác thông thường. Những hàm hyperbolic cơ bản gồm <​b>​sin hyperbolic</​b>​ "​sinh",​ và <​b>​cosin hyperbolic</​b>​ "​cosh",​ hàm <​b>​tang hyperbolic</​b>​ "​tanh"​ và những hàm dẫn ra từ chúng, tương ứng như các hàm dẫn xuất trong hàm lượng giác. ​ Hàm hyperbolic ngược là các hàm <​b>​sin hyperbolic diện tích</​b>​ "​arsinh"​ (hay "​asinh"​ hoặc "​arcsinh"​)<​sup id="​cite_ref-1"​ class="​reference">​[1]</​sup>​.
 +</​p><​p>​Giống như các điểm (cos <​i>​t</​i>,​ sin <​i>​t</​i>​) nằm trên đường tròn bán kính đơn vị, các điểm (cosh <​i>​t</​i>,​ sinh <​i>​t</​i>​) nằm trên phần bên phải của hyperbol đều. Các hàm Hyperbol xuất hiện nhiều trong các nghiệm của các phương trình vi phân tuyến tính hay gặp, phương trình xác định hình dạng dây xích treo giữa 2 điểm, và phương trình Laplace trong hệ tọa độ Descartes. Ngoài ra chúng còn xuất hiện nhiều trong các vấn đề bao gồm lý thuyết điện từ, sự truyền nhiệt, thủy động lực học, và thuyết tương đối hẹp.
 +</​p><​p>​Hàm hyperbolic nhận giá trị thực đối với các tham số thực được gọi là góc hyperbolic. Trong giải tích phức, chúng chính là những hàm mũ hữu tỉ, hay là hàm phân hình (en:​meromorphic function).
 +</​p><​p>​Các hàm hyperbolic được hai nhà toán học Vincenzo Riccati và Johann Heinrich Lambert độc lập đưa ra vào những năm 1760.<​sup id="​cite_ref-2"​ class="​reference">​[2]</​sup>​ Riccati sử dụng ký hiệu <​i>​Sc.</​i>​ và <​i>​Cc.</​i>​ (<​i>​[co]sinus circulare</​i>​) để nói đến các hàm lượng giác <​i>​Sh.</​i>​ và <​i>​Ch.</​i>​ (<​i>​[co]sinus hyperbolico</​i>​) để nói đến các hàm hyperbolic. Lambert là người đã đưa ra các ký hiệu được sử dụng như ngày nay.<sup id="​cite_ref-3"​ class="​reference">​[3]</​sup></​p>​
  
 +
 +<​h2><​span id="​Bi.E1.BB.83u_th.E1.BB.A9c_c.E1.BB.A7a_c.C3.A1c_h.C3.A0m_hyperbolic"/><​span class="​mw-headline"​ id="​Biểu_thức_của_các_hàm_hyperbolic">​Biểu thức của các hàm hyperbolic</​span><​span class="​mw-editsection"><​span class="​mw-editsection-bracket">​[</​span>​sửa<​span class="​mw-editsection-divider">​ | </​span>​sửa mã nguồn<​span class="​mw-editsection-bracket">​]</​span></​span></​h2>​
 +<div class="​thumb tright"><​div class="​thumbinner"​ style="​width:​258px;"><​img alt=""​ src="​http://​upload.wikimedia.org/​wikipedia/​commons/​thumb/​7/​76/​Sinh_cosh_tanh.svg/​256px-Sinh_cosh_tanh.svg.png"​ width="​256"​ height="​256"​ class="​thumbimage"​ srcset="//​upload.wikimedia.org/​wikipedia/​commons/​thumb/​7/​76/​Sinh_cosh_tanh.svg/​384px-Sinh_cosh_tanh.svg.png 1.5x, //​upload.wikimedia.org/​wikipedia/​commons/​thumb/​7/​76/​Sinh_cosh_tanh.svg/​512px-Sinh_cosh_tanh.svg.png 2x" data-file-width="​504"​ data-file-height="​504"/> ​ </​div></​div>​
 +<div class="​thumb tright"><​div class="​thumbinner"​ style="​width:​258px;"><​img alt=""​ src="​http://​upload.wikimedia.org/​wikipedia/​commons/​thumb/​6/​61/​Csch_sech_coth.svg/​256px-Csch_sech_coth.svg.png"​ width="​256"​ height="​256"​ class="​thumbimage"​ srcset="//​upload.wikimedia.org/​wikipedia/​commons/​thumb/​6/​61/​Csch_sech_coth.svg/​384px-Csch_sech_coth.svg.png 1.5x, //​upload.wikimedia.org/​wikipedia/​commons/​thumb/​6/​61/​Csch_sech_coth.svg/​512px-Csch_sech_coth.svg.png 2x" data-file-width="​504"​ data-file-height="​504"/> ​ </​div></​div>​
 +<​p>​Công thức biểu diễn các hàm hyperbolic:
 +</p>
 +<​dl><​dd><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle sinh x={frac {e^{x}-e^{-x}}{2}}={frac {e^{2x}-1}{2e^{x}}}}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mo>​=</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​msup><​mi>​e</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​x</​mi></​mrow></​msup><​mo>​−<​!-- &minus; --></​mo><​msup><​mi>​e</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mo>​−<​!-- &minus; --></​mo><​mi>​x</​mi></​mrow></​msup></​mrow><​mn>​2</​mn></​mfrac></​mrow><​mo>​=</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​msup><​mi>​e</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn><​mi>​x</​mi></​mrow></​msup><​mo>​−<​!-- &minus; --></​mo><​mn>​1</​mn></​mrow><​mrow><​mn>​2</​mn><​msup><​mi>​e</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​x</​mi></​mrow></​msup></​mrow></​mfrac></​mrow></​mstyle></​mrow>​{displaystyle sinh x={frac {e^{x}-e^{-x}}{2}}={frac {e^{2x}-1}{2e^{x}}}}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​6b537c3825615d05ed53e2e95e3f34da3000e570"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -1.838ex; width:​29.446ex;​ height:​5.676ex;"​ alt="​{displaystyle sinh x={frac {e^{x}-e^{-x}}{2}}={frac {e^{2x}-1}{2e^{x}}}}"/></​span></​dd></​dl></​dd></​dl><​dl><​dd><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle cosh x={frac {e^{x}+e^{-x}}{2}}={frac {e^{2x}+1}{2e^{x}}}}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mo>​=</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​msup><​mi>​e</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​x</​mi></​mrow></​msup><​mo>​+</​mo><​msup><​mi>​e</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mo>​−<​!-- &minus; --></​mo><​mi>​x</​mi></​mrow></​msup></​mrow><​mn>​2</​mn></​mfrac></​mrow><​mo>​=</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​msup><​mi>​e</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn><​mi>​x</​mi></​mrow></​msup><​mo>​+</​mo><​mn>​1</​mn></​mrow><​mrow><​mn>​2</​mn><​msup><​mi>​e</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​x</​mi></​mrow></​msup></​mrow></​mfrac></​mrow></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle cosh x={frac {e^{x}+e^{-x}}{2}}={frac {e^{2x}+1}{2e^{x}}}}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​e2acc1509a9e9e5f4c8c89c79468035b2a3f18c0"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -1.838ex; width:​29.701ex;​ height:​5.676ex;"​ alt="​{displaystyle cosh x={frac {e^{x}+e^{-x}}{2}}={frac {e^{2x}+1}{2e^{x}}}}"/></​span></​dd></​dl></​dd></​dl><​dl><​dd><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle tanh x={frac {sinh x}{cosh x}}={frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}={frac {e^{2x}-1}{e^{2x}+1}}}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​tanh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mo>​=</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi></​mrow><​mrow><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi></​mrow></​mfrac></​mrow><​mo>​=</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​msup><​mi>​e</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​x</​mi></​mrow></​msup><​mo>​−<​!-- &minus; --></​mo><​msup><​mi>​e</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mo>​−<​!-- &minus; --></​mo><​mi>​x</​mi></​mrow></​msup></​mrow><​mrow><​msup><​mi>​e</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​x</​mi></​mrow></​msup><​mo>​+</​mo><​msup><​mi>​e</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mo>​−<​!-- &minus; --></​mo><​mi>​x</​mi></​mrow></​msup></​mrow></​mfrac></​mrow><​mo>​=</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​msup><​mi>​e</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn><​mi>​x</​mi></​mrow></​msup><​mo>​−<​!-- &minus; --></​mo><​mn>​1</​mn></​mrow><​mrow><​msup><​mi>​e</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn><​mi>​x</​mi></​mrow></​msup><​mo>​+</​mo><​mn>​1</​mn></​mrow></​mfrac></​mrow></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle tanh x={frac {sinh x}{cosh x}}={frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}={frac {e^{2x}-1}{e^{2x}+1}}}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​3f2cf2340523c38532abdb7103962b9f990701d5"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -2.338ex; width:​40.005ex;​ height:​6.176ex;"​ alt="​{displaystyle tanh x={frac {sinh x}{cosh x}}={frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}={frac {e^{2x}-1}{e^{2x}+1}}}"/></​span></​dd></​dl></​dd></​dl><​dl><​dd><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle coth x={frac {cosh x}{sinh x}}={frac {e^{x}+e^{-x}}{e^{x}-e^{-x}}}={frac {e^{2x}+1}{e^{2x}-1}}}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​coth</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mo>​=</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi></​mrow><​mrow><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi></​mrow></​mfrac></​mrow><​mo>​=</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​msup><​mi>​e</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​x</​mi></​mrow></​msup><​mo>​+</​mo><​msup><​mi>​e</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mo>​−<​!-- &minus; --></​mo><​mi>​x</​mi></​mrow></​msup></​mrow><​mrow><​msup><​mi>​e</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​x</​mi></​mrow></​msup><​mo>​−<​!-- &minus; --></​mo><​msup><​mi>​e</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mo>​−<​!-- &minus; --></​mo><​mi>​x</​mi></​mrow></​msup></​mrow></​mfrac></​mrow><​mo>​=</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​msup><​mi>​e</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn><​mi>​x</​mi></​mrow></​msup><​mo>​+</​mo><​mn>​1</​mn></​mrow><​mrow><​msup><​mi>​e</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn><​mi>​x</​mi></​mrow></​msup><​mo>​−<​!-- &minus; --></​mo><​mn>​1</​mn></​mrow></​mfrac></​mrow></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle coth x={frac {cosh x}{sinh x}}={frac {e^{x}+e^{-x}}{e^{x}-e^{-x}}}={frac {e^{2x}+1}{e^{2x}-1}}}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​b2294b690798596190403626f875384df3767218"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -2.338ex; width:​39.745ex;​ height:​6.176ex;"​ alt="​{displaystyle coth x={frac {cosh x}{sinh x}}={frac {e^{x}+e^{-x}}{e^{x}-e^{-x}}}={frac {e^{2x}+1}{e^{2x}-1}}}"/></​span></​dd></​dl></​dd></​dl><​dl><​dd><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle operatorname {sech} ,​x=left(cosh xright)^{-1}={frac {2}{e^{x}+e^{-x}}}={frac {2e^{x}}{e^{2x}+1}}}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​sech</​mi><​mspace width="​thinmathspace"/><​mi>​x</​mi><​mo>​=</​mo><​msup><​mrow><​mo>​(</​mo><​mrow><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi></​mrow><​mo>​)</​mo></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mo>​−<​!-- &minus; --></​mo><​mn>​1</​mn></​mrow></​msup><​mo>​=</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mn>​2</​mn><​mrow><​msup><​mi>​e</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​x</​mi></​mrow></​msup><​mo>​+</​mo><​msup><​mi>​e</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mo>​−<​!-- &minus; --></​mo><​mi>​x</​mi></​mrow></​msup></​mrow></​mfrac></​mrow><​mo>​=</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mn>​2</​mn><​msup><​mi>​e</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​x</​mi></​mrow></​msup></​mrow><​mrow><​msup><​mi>​e</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn><​mi>​x</​mi></​mrow></​msup><​mo>​+</​mo><​mn>​1</​mn></​mrow></​mfrac></​mrow></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle operatorname {sech} ,​x=left(cosh xright)^{-1}={frac {2}{e^{x}+e^{-x}}}={frac {2e^{x}}{e^{2x}+1}}}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​4db1c54f36a9d4b5b2bfe1db8dbfa448644bfe0a"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -2.338ex; width:​43.319ex;​ height:​5.676ex;"​ alt="​{displaystyle operatorname {sech} ,​x=left(cosh xright)^{-1}={frac {2}{e^{x}+e^{-x}}}={frac {2e^{x}}{e^{2x}+1}}}"/></​span></​dd></​dl></​dd></​dl><​dl><​dd><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle operatorname {csch} ,​x=left(sinh xright)^{-1}={frac {2}{e^{x}-e^{-x}}}={frac {2e^{x}}{e^{2x}-1}}}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​csch</​mi><​mspace width="​thinmathspace"/><​mi>​x</​mi><​mo>​=</​mo><​msup><​mrow><​mo>​(</​mo><​mrow><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi></​mrow><​mo>​)</​mo></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mo>​−<​!-- &minus; --></​mo><​mn>​1</​mn></​mrow></​msup><​mo>​=</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mn>​2</​mn><​mrow><​msup><​mi>​e</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​x</​mi></​mrow></​msup><​mo>​−<​!-- &minus; --></​mo><​msup><​mi>​e</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mo>​−<​!-- &minus; --></​mo><​mi>​x</​mi></​mrow></​msup></​mrow></​mfrac></​mrow><​mo>​=</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mn>​2</​mn><​msup><​mi>​e</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​x</​mi></​mrow></​msup></​mrow><​mrow><​msup><​mi>​e</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn><​mi>​x</​mi></​mrow></​msup><​mo>​−<​!-- &minus; --></​mo><​mn>​1</​mn></​mrow></​mfrac></​mrow></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle operatorname {csch} ,​x=left(sinh xright)^{-1}={frac {2}{e^{x}-e^{-x}}}={frac {2e^{x}}{e^{2x}-1}}}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​4120005e03cbfc4dc43707b1ba3f3adb8788a5fd"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -2.338ex; width:​43.064ex;​ height:​5.676ex;"​ alt="​{displaystyle operatorname {csch} ,​x=left(sinh xright)^{-1}={frac {2}{e^{x}-e^{-x}}}={frac {2e^{x}}{e^{2x}-1}}}"/></​span></​dd></​dl></​dd></​dl><​p>​Các hàm hyperbolic có thể biểu diễn qua <a href="​http://​vi.wikipedia.org/​wiki/​S%E1%BB%91_ph%E1%BB%A9c"​ title="​Số phức">​số phức:
 +</p>
 +<​dl><​dd><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle sinh x=-{rm {i}}sin {rm {i}}x!}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mo>​=</​mo><​mo>​−<​!-- &minus; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​normal">​i</​mi></​mrow></​mrow><​mi>​sin</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​normal">​i</​mi></​mrow></​mrow><​mi>​x</​mi><​mspace width="​negativethinmathspace"/></​mstyle></​mrow>​{displaystyle sinh x=-{rm {i}}sin {rm {i}}x!}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​68aa711998e2c4dfd6983557ba7a6b82d04dd17b"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -0.505ex; margin-right:​ -0.271ex; width:​16.909ex;​ height:​2.343ex;"​ alt="​{displaystyle sinh x=-{rm {i}}sin {rm {i}}x!}"/></​span></​dd></​dl></​dd></​dl><​dl><​dd><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle cosh x=cos {rm {i}}x!}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mo>​=</​mo><​mi>​cos</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​normal">​i</​mi></​mrow></​mrow><​mi>​x</​mi><​mspace width="​negativethinmathspace"/></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle cosh x=cos {rm {i}}x!}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​1ae90468a74344a1f91c3a43861685f68105eed3"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -0.338ex; margin-right:​ -0.271ex; width:​14.577ex;​ height:​2.176ex;"​ alt="​{displaystyle cosh x=cos {rm {i}}x!}"/></​span></​dd></​dl></​dd></​dl><​dl><​dd><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle tanh x=-{rm {i}}tan {rm {i}}x!}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​tanh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mo>​=</​mo><​mo>​−<​!-- &minus; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​normal">​i</​mi></​mrow></​mrow><​mi>​tan</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​normal">​i</​mi></​mrow></​mrow><​mi>​x</​mi><​mspace width="​negativethinmathspace"/></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle tanh x=-{rm {i}}tan {rm {i}}x!}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​9bd378c06c3ef23629255e1d0690552120d200ff"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -0.505ex; margin-right:​ -0.271ex; width:​17.917ex;​ height:​2.343ex;"​ alt="​{displaystyle tanh x=-{rm {i}}tan {rm {i}}x!}"/></​span></​dd></​dl></​dd></​dl><​dl><​dd><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle coth x={rm {i}}cot {rm {i}}x!}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​coth</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mo>​=</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​normal">​i</​mi></​mrow></​mrow><​mi>​cot</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​normal">​i</​mi></​mrow></​mrow><​mi>​x</​mi><​mspace width="​negativethinmathspace"/></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle coth x={rm {i}}cot {rm {i}}x!}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​744a686bcfced1b0a13d6dd3ae86c7b2ea5c6f10"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -0.338ex; margin-right:​ -0.271ex; width:​15.588ex;​ height:​2.176ex;"​ alt="​{displaystyle coth x={rm {i}}cot {rm {i}}x!}"/></​span></​dd></​dl></​dd></​dl><​dl><​dd><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle operatorname {sech} ,x=sec {{rm {i}}x}!}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​sech</​mi><​mspace width="​thinmathspace"/><​mi>​x</​mi><​mo>​=</​mo><​mi>​sec</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mrow class="​MJX-TeXAtom-ORD"><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​normal">​i</​mi></​mrow></​mrow><​mi>​x</​mi></​mrow><​mspace width="​negativethinmathspace"/></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle operatorname {sech} ,x=sec {{rm {i}}x}!}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​764110831aa8cf345a59d018f39c3f43cec8b2e1"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -0.338ex; margin-right:​ -0.271ex; width:​14.704ex;​ height:​2.176ex;"​ alt="​{displaystyle operatorname {sech} ,x=sec {{rm {i}}x}!}"/></​span></​dd></​dl></​dd></​dl><​dl><​dd><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle operatorname {csch} ,x={rm {i}},csc ,{rm {i}}x!}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​csch</​mi><​mspace width="​thinmathspace"/><​mi>​x</​mi><​mo>​=</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​normal">​i</​mi></​mrow></​mrow><​mspace width="​thinmathspace"/><​mi>​csc</​mi><​mspace width="​thinmathspace"/><​mrow class="​MJX-TeXAtom-ORD"><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​normal">​i</​mi></​mrow></​mrow><​mi>​x</​mi><​mspace width="​negativethinmathspace"/></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle operatorname {csch} ,x={rm {i}},csc ,{rm {i}}x!}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​90ec759a65df9c24c4b96fe4d9a5d25142416aae"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -0.338ex; margin-right:​ -0.271ex; width:​16.513ex;​ height:​2.176ex;"​ alt="​{displaystyle operatorname {csch} ,x={rm {i}},csc ,{rm {i}}x!}"/></​span></​dd></​dl></​dd></​dl><​p>​với <​i>​i</​i>​ là <a href="​http://​vi.wikipedia.org/​wiki/​%C4%90%C6%A1n_v%E1%BB%8B_%E1%BA%A3o"​ title="​Đơn vị ảo">​đơn vị ảo định nghĩa là <​i>​i</​i><​sup>​2</​sup>​ = −1.
 +</​p><​p>​Dạng phức trong các định nghĩa trên được dẫn ra từ công thức Euler.
 +</​p><​p>​Chú ý rằng, theo định nghĩa, sinh<​sup>​2</​sup><​i>​x</​i>​ có nghĩa là (sinh <​i>​x</​i>​)<​sup>​2</​sup>,​ chứ không phải sinh(sinh <​i>​x</​i>​);​ và điều này tương tự cho các hàm hyperbolic khác.
 +</p>
 +<​h2><​span id="​M.E1.BB.91i_quan_h.E1.BB.87_gi.E1.BB.AFa_c.C3.A1c_h.C3.A0m_hyperbolic"/><​span class="​mw-headline"​ id="​Mối_quan_hệ_giữa_các_hàm_hyperbolic">​Mối quan hệ giữa các hàm hyperbolic</​span><​span class="​mw-editsection"><​span class="​mw-editsection-bracket">​[</​span>​sửa<​span class="​mw-editsection-divider">​ | </​span>​sửa mã nguồn<​span class="​mw-editsection-bracket">​]</​span></​span></​h2>​
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle sinh(-x)=-sinh x,​!}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mo>​−<​!-- &minus; --></​mo><​mi>​x</​mi><​mo stretchy="​false">​)</​mo><​mo>​=</​mo><​mo>​−<​!-- &minus; --></​mo><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mspace width="​thinmathspace"/><​mspace width="​negativethinmathspace"/></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle sinh(-x)=-sinh x,​!}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​e3d142a267964eb67e6fc7bec54f3269ce22a151"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -0.838ex; margin-right:​ -0.387ex; width:​20.641ex;​ height:​2.843ex;"​ alt="​{displaystyle sinh(-x)=-sinh x,​!}"/></​span></​dd>​
 +<​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle cosh(-x)=cosh x,​!}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mo>​−<​!-- &minus; --></​mo><​mi>​x</​mi><​mo stretchy="​false">​)</​mo><​mo>​=</​mo><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mspace width="​thinmathspace"/><​mspace width="​negativethinmathspace"/></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle cosh(-x)=cosh x,​!}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​7d71e0752cee526572ae1f0723a8e539a84316f8"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -0.838ex; margin-right:​ -0.387ex; width:​18.957ex;​ height:​2.843ex;"​ alt="​{displaystyle cosh(-x)=cosh x,​!}"/></​span></​dd></​dl><​p>​Từ đó:
 +</p>
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle tanh(-x)=-tanh x,​!}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​tanh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mo>​−<​!-- &minus; --></​mo><​mi>​x</​mi><​mo stretchy="​false">​)</​mo><​mo>​=</​mo><​mo>​−<​!-- &minus; --></​mo><​mi>​tanh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mspace width="​thinmathspace"/><​mspace width="​negativethinmathspace"/></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle tanh(-x)=-tanh x,​!}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​b48cd09fe94e7f8bdeea8d9f6ce127f868311231"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -0.838ex; margin-right:​ -0.387ex; width:​21.649ex;​ height:​2.843ex;"​ alt="​{displaystyle tanh(-x)=-tanh x,​!}"/></​span></​dd>​
 +<​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle coth(-x)=-coth x,​!}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​coth</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mo>​−<​!-- &minus; --></​mo><​mi>​x</​mi><​mo stretchy="​false">​)</​mo><​mo>​=</​mo><​mo>​−<​!-- &minus; --></​mo><​mi>​coth</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mspace width="​thinmathspace"/><​mspace width="​negativethinmathspace"/></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle coth(-x)=-coth x,​!}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​8743c282c6a72cdaf9c742e56f53aeca45b4204d"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -0.838ex; margin-right:​ -0.387ex; width:​21.129ex;​ height:​2.843ex;"​ alt="​{displaystyle coth(-x)=-coth x,​!}"/></​span></​dd>​
 +<​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle operatorname {sech} (-x)=operatorname {sech} ,​x,​!}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​sech</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mo>​−<​!-- &minus; --></​mo><​mi>​x</​mi><​mo stretchy="​false">​)</​mo><​mo>​=</​mo><​mi>​sech</​mi><​mspace width="​thinmathspace"/><​mi>​x</​mi><​mspace width="​thinmathspace"/><​mspace width="​negativethinmathspace"/></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle operatorname {sech} (-x)=operatorname {sech} ,​x,​!}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​eca2f63a1c8b204a40660f6e865591101c3131eb"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -0.838ex; margin-right:​ -0.387ex; width:​19.084ex;​ height:​2.843ex;"​ alt="​{displaystyle operatorname {sech} (-x)=operatorname {sech} ,​x,​!}"/></​span></​dd>​
 +<​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle operatorname {csch} (-x)=-operatorname {csch} ,​x,​!}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​csch</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mo>​−<​!-- &minus; --></​mo><​mi>​x</​mi><​mo stretchy="​false">​)</​mo><​mo>​=</​mo><​mo>​−<​!-- &minus; --></​mo><​mi>​csch</​mi><​mspace width="​thinmathspace"/><​mi>​x</​mi><​mspace width="​thinmathspace"/><​mspace width="​negativethinmathspace"/></​mstyle></​mrow>​{displaystyle operatorname {csch} (-x)=-operatorname {csch} ,​x,​!}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​9735edc0c3a950f62e5a611bf6d96daa17e5dfa5"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -0.838ex; margin-right:​ -0.387ex; width:​21.279ex;​ height:​2.843ex;"​ alt="​{displaystyle operatorname {csch} (-x)=-operatorname {csch} ,​x,​!}"/></​span></​dd></​dl><​p>​Theo quan hệ trên dễ thấy cosh <​i>​x</​i>​ và sech <​i>​x</​i>​ là các <a href="​http://​vi.wikipedia.org/​w/​index.php?​title=H%C3%A0m_ch%E1%BA%B5n&​amp;​action=edit&​amp;​redlink=1"​ class="​new"​ title="​Hàm chẵn (trang chưa được viết)">​hàm chẵn; còn lại là các hàm lẻ.
 +</p>
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle operatorname {arsech} ,​x=operatorname {arcosh} {frac {1}{x}}}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​arsech</​mi><​mspace width="​thinmathspace"/><​mi>​x</​mi><​mo>​=</​mo><​mi>​arcosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mn>​1</​mn><​mi>​x</​mi></​mfrac></​mrow></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle operatorname {arsech} ,​x=operatorname {arcosh} {frac {1}{x}}}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​8cab42cbddc1c2e5bbd771cc9a8a2dfd72f40821"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -1.838ex; width:​20.581ex;​ height:​5.176ex;"​ alt="​{displaystyle operatorname {arsech} ,​x=operatorname {arcosh} {frac {1}{x}}}"/></​span></​dd></​dl><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle operatorname {arcsch} ,​x=operatorname {arsinh} {frac {1}{x}}}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​arcsch</​mi><​mspace width="​thinmathspace"/><​mi>​x</​mi><​mo>​=</​mo><​mi>​arsinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mn>​1</​mn><​mi>​x</​mi></​mfrac></​mrow></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle operatorname {arcsch} ,​x=operatorname {arsinh} {frac {1}{x}}}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​01de1b54e5f112079e41dc9dbd59a1c64a7d4ddb"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -1.838ex; width:​20.325ex;​ height:​5.176ex;"​ alt="​{displaystyle operatorname {arcsch} ,​x=operatorname {arsinh} {frac {1}{x}}}"/></​span></​dd></​dl><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle operatorname {arcoth} ,​x=operatorname {artanh} {frac {1}{x}}}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​arcoth</​mi><​mspace width="​thinmathspace"/><​mi>​x</​mi><​mo>​=</​mo><​mi>​artanh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mn>​1</​mn><​mi>​x</​mi></​mfrac></​mrow></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle operatorname {arcoth} ,​x=operatorname {artanh} {frac {1}{x}}}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​9d21595b05425a11d7ef52bb04392d262e5fc491"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -1.838ex; width:​20.948ex;​ height:​5.176ex;"​ alt="​{displaystyle operatorname {arcoth} ,​x=operatorname {artanh} {frac {1}{x}}}"/></​span></​dd></​dl><​p>​Sin hyperbolic và cos hyperbolic thỏa mãn đẳng thức
 +</p>
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle cosh ^{2}x-sinh ^{2}x=1,​}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​msup><​mi>​cosh</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mo>​−<​!-- &minus; --></​mo><​msup><​mi>​sinh</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mo>​=</​mo><​mn>​1</​mn><​mspace width="​thinmathspace"/></​mstyle></​mrow>​{displaystyle cosh ^{2}x-sinh ^{2}x=1,​}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​b55561450defdca67aa5f2ed44a01d5f3810d805"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -0.505ex; width:​21.582ex;​ height:​2.843ex;"​ alt="​{displaystyle cosh ^{2}x-sinh ^{2}x=1,​}"/></​span></​dd></​dl><​p>​tương tự như <a href="​http://​vi.wikipedia.org/​w/​index.php?​title=C%C3%B4ng_th%E1%BB%A9c_l%C6%B0%E1%BB%A3ng_gi%C3%A1c_Pythagore&​amp;​action=edit&​amp;​redlink=1"​ class="​new"​ title="​Công thức lượng giác Pythagore (trang chưa được viết)">​công thức lượng giác Pythagore: ​
 +<span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle sin ^{2}theta +cos ^{2}theta =1.!}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​msup><​mi>​sin</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​θ<​!-- &theta; --></​mi><​mo>​+</​mo><​msup><​mi>​cos</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​θ<​!-- &theta; --></​mi><​mo>​=</​mo><​mn>​1.</​mn><​mspace width="​negativethinmathspace"/></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle sin ^{2}theta +cos ^{2}theta =1.!}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​fb2a52f40159bb3b6209e05813ddfbf80dad1c18"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -0.505ex; margin-right:​ -0.204ex; width:​18.595ex;​ height:​2.843ex;"​ alt="​{displaystyle sin ^{2}theta +cos ^{2}theta =1.!}"/></​span>​. ​
 +Do vậy ta cũng có:
 +</p>
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle tanh ^{2}x=1-operatorname {sech} ^{2}x}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​msup><​mi>​tanh</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mo>​=</​mo><​mn>​1</​mn><​mo>​−<​!-- &minus; --></​mo><​msup><​mi>​sech</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle tanh ^{2}x=1-operatorname {sech} ^{2}x}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​7f90836e864404f04c6f8528de4972b1d4ba2efa"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -0.505ex; width:​21.569ex;​ height:​2.843ex;"​ alt="​{displaystyle tanh ^{2}x=1-operatorname {sech} ^{2}x}"/></​span></​dd>​
 +<​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle coth ^{2}x=1+operatorname {csch} ^{2}x}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​msup><​mi>​coth</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mo>​=</​mo><​mn>​1</​mn><​mo>​+</​mo><​msup><​mi>​csch</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi></​mstyle></​mrow>​{displaystyle coth ^{2}x=1+operatorname {csch} ^{2}x}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​cd9ddaa029c3f8da2cd33dfc0111e8acff66f434"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -0.505ex; width:​21.309ex;​ height:​2.843ex;"​ alt="​{displaystyle coth ^{2}x=1+operatorname {csch} ^{2}x}"/></​span></​dd></​dl><​p>​Tang hyperbolic là nghiệm của <a href="​http://​vi.wikipedia.org/​w/​index.php?​title=B%C3%A0i_to%C3%A1n_gi%C3%A1_tr%E1%BB%8B_bi%C3%AAn&​amp;​action=edit&​amp;​redlink=1"​ class="​new"​ title="​Bài toán giá trị biên (trang chưa được viết)">​bài toán giá trị biên phi tuyến<​sup id="​cite_ref-4"​ class="​reference">​[4]</​sup>:​
 +</p>
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle {frac {1}{2}}f''​=f^{3}-fqquad ;qquad f(0)=f'​(infty )=0}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mn>​1</​mn><​mn>​2</​mn></​mfrac></​mrow><​msup><​mi>​f</​mi><​mo>​″</​mo></​msup><​mo>​=</​mo><​msup><​mi>​f</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​3</​mn></​mrow></​msup><​mo>​−<​!-- &minus; --></​mo><​mi>​f</​mi><​mspace width="​2em"/><​mo>;</​mo><​mspace width="​2em"/><​mi>​f</​mi><​mo stretchy="​false">​(</​mo><​mn>​0</​mn><​mo stretchy="​false">​)</​mo><​mo>​=</​mo><​msup><​mi>​f</​mi><​mo>​′</​mo></​msup><​mo stretchy="​false">​(</​mo><​mi mathvariant="​normal">​∞<​!-- &infin; --></​mi><​mo stretchy="​false">​)</​mo><​mo>​=</​mo><​mn>​0</​mn></​mstyle></​mrow>​{displaystyle {frac {1}{2}}f''​=f^{3}-fqquad ;qquad f(0)=f'​(infty )=0}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​c1caa85f79ad670a74341702c6237db74053963e"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -1.838ex; width:​42.121ex;​ height:​5.176ex;"​ alt="​{displaystyle {frac {1}{2}}f''​=f^{3}-fqquad ;qquad f(0)=f'​(infty )=0}"/></​span></​dd></​dl><​p>​Người ta đã chứng minh rằng diện tích giới hạn bởi cung cosh <​i>​x</​i>​ luôn luôn bằng chiều dài của cung đó:<​sup id="​cite_ref-5"​ class="​reference"><​a href="#​cite_note-5">​[5]</​sup></​p>​
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle {text{dien tich}}=int _{a}^{b}{cosh {x}} dx=int _{a}^{b}{sqrt {1+left({frac {d}{dx}}cosh {x}right)^{2}}} dx={text{do dai cung}}.}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mrow class="​MJX-TeXAtom-ORD"><​mtext>​dien tich</​mtext></​mrow><​mo>​=</​mo><​msubsup><​mo>​∫<​!-- &int; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi>​a</​mi></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi>​b</​mi></​mrow></​msubsup><​mrow class="​MJX-TeXAtom-ORD"><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi>​x</​mi></​mrow></​mrow><​mtext>​ </​mtext><​mi>​d</​mi><​mi>​x</​mi><​mo>​=</​mo><​msubsup><​mo>​∫<​!-- &int; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi>​a</​mi></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi>​b</​mi></​mrow></​msubsup><​mrow class="​MJX-TeXAtom-ORD"><​msqrt><​mn>​1</​mn><​mo>​+</​mo><​msup><​mrow><​mo>​(</​mo><​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mi>​d</​mi><​mrow><​mi>​d</​mi><​mi>​x</​mi></​mrow></​mfrac></​mrow><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi>​x</​mi></​mrow></​mrow><​mo>​)</​mo></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup></​msqrt></​mrow><​mtext>​ </​mtext><​mi>​d</​mi><​mi>​x</​mi><​mo>​=</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mtext>​do dai cung</​mtext></​mrow><​mo>​.</​mo></​mstyle></​mrow>​{displaystyle {text{dien tich}}=int _{a}^{b}{cosh {x}} dx=int _{a}^{b}{sqrt {1+left({frac {d}{dx}}cosh {x}right)^{2}}} dx={text{do dai cung}}.}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​d49bad6b938eeae25b04a6cc50d1fe97de796656"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -3.005ex; width:​70.803ex;​ height:​7.676ex;"​ alt="​{displaystyle {text{dien tich}}=int _{a}^{b}{cosh {x}} dx=int _{a}^{b}{sqrt {1+left({frac {d}{dx}}cosh {x}right)^{2}}} dx={text{do dai cung}}.}"/></​span></​dd></​dl><​h3><​span id="​C.E1.BB.99ng_c.C3.A1c_.C4.91.E1.BB.91i_s.E1.BB.91"/><​span class="​mw-headline"​ id="​Cộng_các_đối_số">​Cộng các đối số</​span><​span class="​mw-editsection"><​span class="​mw-editsection-bracket">​[</​span><​a href="​http://​vi.wikipedia.org/​w/​index.php?​title=H%C3%A0m_hypebolic&​amp;​veaction=edit&​amp;​section=3"​ class="​mw-editsection-visualeditor"​ title="​Sửa đổi phần “Cộng các đối số”">​sửa<​span class="​mw-editsection-divider">​ | </​span>​sửa mã nguồn<​span class="​mw-editsection-bracket">​]</​span></​span></​h3>​
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle {begin{aligned}sinh(x+y)&​amp;​=sinh(x)cosh(y)+cosh(x)sinh(y)\cosh(x+y)&​amp;​=cosh(x)cosh(y)+sinh(x)sinh(y)\tanh(x+y)&​amp;​={frac {tanh x+tanh y}{1+tanh xtanh y}}\end{aligned}}}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mrow class="​MJX-TeXAtom-ORD"><​mtable columnalign="​right left right left right left right left right left right left" rowspacing="​3pt"​ columnspacing="​0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="​true"><​mtr><​mtd><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​x</​mi><​mo>​+</​mo><​mi>​y</​mi><​mo stretchy="​false">​)</​mo></​mtd><​mtd><​mi/><​mo>​=</​mo><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​x</​mi><​mo stretchy="​false">​)</​mo><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​y</​mi><​mo stretchy="​false">​)</​mo><​mo>​+</​mo><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​x</​mi><​mo stretchy="​false">​)</​mo><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​y</​mi><​mo stretchy="​false">​)</​mo></​mtd></​mtr><​mtr><​mtd><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​x</​mi><​mo>​+</​mo><​mi>​y</​mi><​mo stretchy="​false">​)</​mo></​mtd><​mtd><​mi/><​mo>​=</​mo><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​x</​mi><​mo stretchy="​false">​)</​mo><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​y</​mi><​mo stretchy="​false">​)</​mo><​mo>​+</​mo><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​x</​mi><​mo stretchy="​false">​)</​mo><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​y</​mi><​mo stretchy="​false">​)</​mo></​mtd></​mtr><​mtr><​mtd><​mi>​tanh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​x</​mi><​mo>​+</​mo><​mi>​y</​mi><​mo stretchy="​false">​)</​mo></​mtd><​mtd><​mi/><​mo>​=</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mi>​tanh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mo>​+</​mo><​mi>​tanh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​y</​mi></​mrow><​mrow><​mn>​1</​mn><​mo>​+</​mo><​mi>​tanh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mi>​tanh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​y</​mi></​mrow></​mfrac></​mrow></​mtd></​mtr></​mtable></​mrow></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle {begin{aligned}sinh(x+y)&​amp;​=sinh(x)cosh(y)+cosh(x)sinh(y)\cosh(x+y)&​amp;​=cosh(x)cosh(y)+sinh(x)sinh(y)\tanh(x+y)&​amp;​={frac {tanh x+tanh y}{1+tanh xtanh y}}\end{aligned}}}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​6ee16379ed3a36460694378808fa49f000f4e1e0"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -5.505ex; width:​48.563ex;​ height:​12.176ex;"​ alt="​{displaystyle {begin{aligned}sinh(x+y)&​amp;​=sinh(x)cosh(y)+cosh(x)sinh(y)\cosh(x+y)&​amp;​=cosh(x)cosh(y)+sinh(x)sinh(y)\tanh(x+y)&​amp;​={frac {tanh x+tanh y}{1+tanh xtanh y}}\end{aligned}}}"/></​span></​dd></​dl><​p>​đặc biệt
 +</p>
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle {begin{aligned}cosh(2x)&​amp;​=sinh ^{2}{x}+cosh ^{2}{x}=2sinh ^{2}x+1=2cosh ^{2}x-1\sinh(2x)&​amp;​=2sinh xcosh xend{aligned}}}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mrow class="​MJX-TeXAtom-ORD"><​mtable columnalign="​right left right left right left right left right left right left" rowspacing="​3pt"​ columnspacing="​0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="​true"><​mtr><​mtd><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mn>​2</​mn><​mi>​x</​mi><​mo stretchy="​false">​)</​mo></​mtd><​mtd><​mi/><​mo>​=</​mo><​msup><​mi>​sinh</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup><​mo>​⁡<​!-- &#8289; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi>​x</​mi></​mrow><​mo>​+</​mo><​msup><​mi>​cosh</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup><​mo>​⁡<​!-- &#8289; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi>​x</​mi></​mrow><​mo>​=</​mo><​mn>​2</​mn><​msup><​mi>​sinh</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mo>​+</​mo><​mn>​1</​mn><​mo>​=</​mo><​mn>​2</​mn><​msup><​mi>​cosh</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mo>​−<​!-- &minus; --></​mo><​mn>​1</​mn></​mtd></​mtr><​mtr><​mtd><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mn>​2</​mn><​mi>​x</​mi><​mo stretchy="​false">​)</​mo></​mtd><​mtd><​mi/><​mo>​=</​mo><​mn>​2</​mn><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi></​mtd></​mtr></​mtable></​mrow></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle {begin{aligned}cosh(2x)&​amp;​=sinh ^{2}{x}+cosh ^{2}{x}=2sinh ^{2}x+1=2cosh ^{2}x-1\sinh(2x)&​amp;​=2sinh xcosh xend{aligned}}}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​ca852b07b1e4fa7155fdb9dbf65922249b887d11"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -2.505ex; width:​60.885ex;​ height:​6.176ex;"​ alt="​{displaystyle {begin{aligned}cosh(2x)&​amp;​=sinh ^{2}{x}+cosh ^{2}{x}=2sinh ^{2}x+1=2cosh ^{2}x-1\sinh(2x)&​amp;​=2sinh xcosh xend{aligned}}}"/></​span></​dd></​dl><​p>​Và:​
 +</p>
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle {begin{aligned}sinh x+sinh y&​amp;​=2sinh {frac {x+y}{2}}cosh {frac {x-y}{2}}\cosh x+cosh y&​amp;​=2cosh {frac {x+y}{2}}cosh {frac {x-y}{2}}\end{aligned}}}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mrow class="​MJX-TeXAtom-ORD"><​mtable columnalign="​right left right left right left right left right left right left" rowspacing="​3pt"​ columnspacing="​0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="​true"><​mtr><​mtd><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mo>​+</​mo><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​y</​mi></​mtd><​mtd><​mi/><​mo>​=</​mo><​mn>​2</​mn><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mi>​x</​mi><​mo>​+</​mo><​mi>​y</​mi></​mrow><​mn>​2</​mn></​mfrac></​mrow><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mi>​x</​mi><​mo>​−<​!-- &minus; --></​mo><​mi>​y</​mi></​mrow><​mn>​2</​mn></​mfrac></​mrow></​mtd></​mtr><​mtr><​mtd><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mo>​+</​mo><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​y</​mi></​mtd><​mtd><​mi/><​mo>​=</​mo><​mn>​2</​mn><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mi>​x</​mi><​mo>​+</​mo><​mi>​y</​mi></​mrow><​mn>​2</​mn></​mfrac></​mrow><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mi>​x</​mi><​mo>​−<​!-- &minus; --></​mo><​mi>​y</​mi></​mrow><​mn>​2</​mn></​mfrac></​mrow></​mtd></​mtr></​mtable></​mrow></​mstyle></​mrow>​{displaystyle {begin{aligned}sinh x+sinh y&​amp;​=2sinh {frac {x+y}{2}}cosh {frac {x-y}{2}}\cosh x+cosh y&​amp;​=2cosh {frac {x+y}{2}}cosh {frac {x-y}{2}}\end{aligned}}}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​9acc17fd955a8bcfdd1d5df2903039ac7f1b797a"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -4.671ex; width:​42.598ex;​ height:​10.509ex;"​ alt="​{displaystyle {begin{aligned}sinh x+sinh y&​amp;​=2sinh {frac {x+y}{2}}cosh {frac {x-y}{2}}\cosh x+cosh y&​amp;​=2cosh {frac {x+y}{2}}cosh {frac {x-y}{2}}\end{aligned}}}"/></​span></​dd></​dl><​h3><​span id="​C.C3.B4ng_th.E1.BB.A9c_tr.E1.BB.AB"/><​span class="​mw-headline"​ id="​Công_thức_trừ">​Công thức trừ</​span><​span class="​mw-editsection"><​span class="​mw-editsection-bracket">​[</​span><​a href="​http://​vi.wikipedia.org/​w/​index.php?​title=H%C3%A0m_hypebolic&​amp;​veaction=edit&​amp;​section=4"​ class="​mw-editsection-visualeditor"​ title="​Sửa đổi phần “Công thức trừ”">​sửa<​span class="​mw-editsection-divider">​ | </​span>​sửa mã nguồn<​span class="​mw-editsection-bracket">​]</​span></​span></​h3>​
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle {begin{aligned}sinh(x-y)&​amp;​=sinh(x)cosh(y)-cosh(x)sinh(y)\cosh(x-y)&​amp;​=cosh(x)cosh(y)-sinh(x)sinh(y)\end{aligned}}}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mrow class="​MJX-TeXAtom-ORD"><​mtable columnalign="​right left right left right left right left right left right left" rowspacing="​3pt"​ columnspacing="​0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="​true"><​mtr><​mtd><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​x</​mi><​mo>​−<​!-- &minus; --></​mo><​mi>​y</​mi><​mo stretchy="​false">​)</​mo></​mtd><​mtd><​mi/><​mo>​=</​mo><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​x</​mi><​mo stretchy="​false">​)</​mo><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​y</​mi><​mo stretchy="​false">​)</​mo><​mo>​−<​!-- &minus; --></​mo><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​x</​mi><​mo stretchy="​false">​)</​mo><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​y</​mi><​mo stretchy="​false">​)</​mo></​mtd></​mtr><​mtr><​mtd><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​x</​mi><​mo>​−<​!-- &minus; --></​mo><​mi>​y</​mi><​mo stretchy="​false">​)</​mo></​mtd><​mtd><​mi/><​mo>​=</​mo><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​x</​mi><​mo stretchy="​false">​)</​mo><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​y</​mi><​mo stretchy="​false">​)</​mo><​mo>​−<​!-- &minus; --></​mo><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​x</​mi><​mo stretchy="​false">​)</​mo><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​y</​mi><​mo stretchy="​false">​)</​mo></​mtd></​mtr></​mtable></​mrow></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle {begin{aligned}sinh(x-y)&​amp;​=sinh(x)cosh(y)-cosh(x)sinh(y)\cosh(x-y)&​amp;​=cosh(x)cosh(y)-sinh(x)sinh(y)\end{aligned}}}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​30260fae22446eae75fb15ff14ca3c7b57e8e42d"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -2.505ex; width:​48.314ex;​ height:​6.176ex;"​ alt="​{displaystyle {begin{aligned}sinh(x-y)&​amp;​=sinh(x)cosh(y)-cosh(x)sinh(y)\cosh(x-y)&​amp;​=cosh(x)cosh(y)-sinh(x)sinh(y)\end{aligned}}}"/></​span></​dd></​dl><​p>​Và:​
 +</p>
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle {begin{aligned}sinh x-sinh y&​amp;​=2cosh {frac {x+y}{2}}sinh {frac {x-y}{2}}\cosh x-cosh y&​amp;​=2sinh {frac {x+y}{2}}sinh {frac {x-y}{2}}\end{aligned}}}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mrow class="​MJX-TeXAtom-ORD"><​mtable columnalign="​right left right left right left right left right left right left" rowspacing="​3pt"​ columnspacing="​0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="​true"><​mtr><​mtd><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mo>​−<​!-- &minus; --></​mo><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​y</​mi></​mtd><​mtd><​mi/><​mo>​=</​mo><​mn>​2</​mn><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mi>​x</​mi><​mo>​+</​mo><​mi>​y</​mi></​mrow><​mn>​2</​mn></​mfrac></​mrow><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mi>​x</​mi><​mo>​−<​!-- &minus; --></​mo><​mi>​y</​mi></​mrow><​mn>​2</​mn></​mfrac></​mrow></​mtd></​mtr><​mtr><​mtd><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mo>​−<​!-- &minus; --></​mo><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​y</​mi></​mtd><​mtd><​mi/><​mo>​=</​mo><​mn>​2</​mn><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mi>​x</​mi><​mo>​+</​mo><​mi>​y</​mi></​mrow><​mn>​2</​mn></​mfrac></​mrow><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mi>​x</​mi><​mo>​−<​!-- &minus; --></​mo><​mi>​y</​mi></​mrow><​mn>​2</​mn></​mfrac></​mrow></​mtd></​mtr></​mtable></​mrow></​mstyle></​mrow>​{displaystyle {begin{aligned}sinh x-sinh y&​amp;​=2cosh {frac {x+y}{2}}sinh {frac {x-y}{2}}\cosh x-cosh y&​amp;​=2sinh {frac {x+y}{2}}sinh {frac {x-y}{2}}\end{aligned}}}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​90a1a6a2e5026c2ac3762ece0eb33907d3e989b4"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -4.671ex; width:​42.343ex;​ height:​10.509ex;"​ alt="​{displaystyle {begin{aligned}sinh x-sinh y&​amp;​=2cosh {frac {x+y}{2}}sinh {frac {x-y}{2}}\cosh x-cosh y&​amp;​=2sinh {frac {x+y}{2}}sinh {frac {x-y}{2}}\end{aligned}}}"/></​span></​dd></​dl><​p>​Nguồn tham khảo.<​sup id="​cite_ref-6"​ class="​reference"><​a href="#​cite_note-6">​[6]</​sup></​p>​
 +<​h3><​span id="​C.C3.B4ng_th.E1.BB.A9c_t.C3.ADnh_m.E1.BB.99t_n.E1.BB.ADa_.C4.91.E1.BB.91i_s.E1.BB.91"/><​span class="​mw-headline"​ id="​Công_thức_tính_một_nửa_đối_số">​Công thức tính một nửa đối số</​span><​span class="​mw-editsection"><​span class="​mw-editsection-bracket">​[</​span>​sửa<​span class="​mw-editsection-divider">​ | </​span>​sửa mã nguồn<​span class="​mw-editsection-bracket">​]</​span></​span></​h3>​
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle sinh left({frac {x}{2}}right)={frac {sinh(x)}{sqrt {2(cosh(x)+1)}}}=operatorname {sgn} (x),{sqrt {frac {cosh(x)-1}{2}}}}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mrow><​mo>​(</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mi>​x</​mi><​mn>​2</​mn></​mfrac></​mrow><​mo>​)</​mo></​mrow><​mo>​=</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​x</​mi><​mo stretchy="​false">​)</​mo></​mrow><​msqrt><​mn>​2</​mn><​mo stretchy="​false">​(</​mo><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​x</​mi><​mo stretchy="​false">​)</​mo><​mo>​+</​mo><​mn>​1</​mn><​mo stretchy="​false">​)</​mo></​msqrt></​mfrac></​mrow><​mo>​=</​mo><​mi>​sgn</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​x</​mi><​mo stretchy="​false">​)</​mo><​mspace width="​thinmathspace"/><​mrow class="​MJX-TeXAtom-ORD"><​msqrt><​mfrac><​mrow><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​x</​mi><​mo stretchy="​false">​)</​mo><​mo>​−<​!-- &minus; --></​mo><​mn>​1</​mn></​mrow><​mn>​2</​mn></​mfrac></​msqrt></​mrow></​mstyle></​mrow>​{displaystyle sinh left({frac {x}{2}}right)={frac {sinh(x)}{sqrt {2(cosh(x)+1)}}}=operatorname {sgn} (x),{sqrt {frac {cosh(x)-1}{2}}}}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​6d8d69246ca77b972e3904724d45913e44911e72"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -3.171ex; width:​54.566ex;​ height:​8.009ex;"​ alt="​{displaystyle sinh left({frac {x}{2}}right)={frac {sinh(x)}{sqrt {2(cosh(x)+1)}}}=operatorname {sgn}(x),​{sqrt {frac {cosh(x)-1}{2}}}}"/></​span></​dd></​dl><​p>​với <​i>​sgn</​i>​ là <a href="​http://​vi.wikipedia.org/​wiki/​Signum"​ title="​Signum">​hàm dấu.
 +</p>
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle cosh left({frac {x}{2}}right)={sqrt {frac {cosh(x)+1}{2}}}}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mrow><​mo>​(</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mi>​x</​mi><​mn>​2</​mn></​mfrac></​mrow><​mo>​)</​mo></​mrow><​mo>​=</​mo><​mrow class="​MJX-TeXAtom-ORD"><​msqrt><​mfrac><​mrow><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​x</​mi><​mo stretchy="​false">​)</​mo><​mo>​+</​mo><​mn>​1</​mn></​mrow><​mn>​2</​mn></​mfrac></​msqrt></​mrow></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle cosh left({frac {x}{2}}right)={sqrt {frac {cosh(x)+1}{2}}}}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​8570fd1100eca6d8759541b618e385072c98bf9b"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -2.838ex; width:​27.149ex;​ height:​7.676ex;"​ alt="​{displaystyle cosh left({frac {x}{2}}right)={sqrt {frac {cosh(x)+1}{2}}}}"/></​span></​dd></​dl><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle tanh left({frac {x}{2}}right)={frac {sinh(x)}{cosh(x)+1}}=operatorname {sgn} (x),{sqrt {frac {cosh(x)-1}{cosh(x)+1}}}={frac {e^{x}-1}{e^{x}+1}}}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​tanh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mrow><​mo>​(</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mi>​x</​mi><​mn>​2</​mn></​mfrac></​mrow><​mo>​)</​mo></​mrow><​mo>​=</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​x</​mi><​mo stretchy="​false">​)</​mo></​mrow><​mrow><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​x</​mi><​mo stretchy="​false">​)</​mo><​mo>​+</​mo><​mn>​1</​mn></​mrow></​mfrac></​mrow><​mo>​=</​mo><​mi>​sgn</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​x</​mi><​mo stretchy="​false">​)</​mo><​mspace width="​thinmathspace"/><​mrow class="​MJX-TeXAtom-ORD"><​msqrt><​mfrac><​mrow><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​x</​mi><​mo stretchy="​false">​)</​mo><​mo>​−<​!-- &minus; --></​mo><​mn>​1</​mn></​mrow><​mrow><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​x</​mi><​mo stretchy="​false">​)</​mo><​mo>​+</​mo><​mn>​1</​mn></​mrow></​mfrac></​msqrt></​mrow><​mo>​=</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​msup><​mi>​e</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​x</​mi></​mrow></​msup><​mo>​−<​!-- &minus; --></​mo><​mn>​1</​mn></​mrow><​mrow><​msup><​mi>​e</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​x</​mi></​mrow></​msup><​mo>​+</​mo><​mn>​1</​mn></​mrow></​mfrac></​mrow></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle tanh left({frac {x}{2}}right)={frac {sinh(x)}{cosh(x)+1}}=operatorname {sgn} (x),{sqrt {frac {cosh(x)-1}{cosh(x)+1}}}={frac {e^{x}-1}{e^{x}+1}}}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​4b4646f605d3b71acfb8c37c6209dd35b267e689"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -3.171ex; width:​59.968ex;​ height:​7.509ex;"​ alt="​{displaystyle tanh left({frac {x}{2}}right)={frac {sinh(x)}{cosh(x)+1}}=operatorname {sgn}(x),​{sqrt {frac {cosh(x)-1}{cosh(x)+1}}}={frac {e^{x}-1}{e^{x}+1}}}"/></​span></​dd></​dl><​p>​Nếu <​i>​x</​i>​ ≠ 0, thì
 +</p>
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle tanh left({frac {x}{2}}right)={frac {cosh(x)-1}{sinh(x)}}=coth(x)-operatorname {csch} (x)}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​tanh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mrow><​mo>​(</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mi>​x</​mi><​mn>​2</​mn></​mfrac></​mrow><​mo>​)</​mo></​mrow><​mo>​=</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​x</​mi><​mo stretchy="​false">​)</​mo><​mo>​−<​!-- &minus; --></​mo><​mn>​1</​mn></​mrow><​mrow><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​x</​mi><​mo stretchy="​false">​)</​mo></​mrow></​mfrac></​mrow><​mo>​=</​mo><​mi>​coth</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​x</​mi><​mo stretchy="​false">​)</​mo><​mo>​−<​!-- &minus; --></​mo><​mi>​csch</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​x</​mi><​mo stretchy="​false">​)</​mo></​mstyle></​mrow>​{displaystyle tanh left({frac {x}{2}}right)={frac {cosh(x)-1}{sinh(x)}}=coth(x)-operatorname {csch} (x)}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​580c34a89e8fb079a7d013183c04bb5e2f15baf5"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -2.671ex; width:​45.956ex;​ height:​6.509ex;"​ alt="​{displaystyle tanh left({frac {x}{2}}right)={frac {cosh(x)-1}{sinh(x)}}=coth(x)-operatorname {csch} (x)}"/></​span><​sup id="​cite_ref-7"​ class="​reference"><​a href="#​cite_note-7">​[7]</​sup></​dd></​dl>​
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle operatorname {arsinh} ,x=ln left(x+{sqrt {x^{2}+1}}right)}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​arsinh</​mi><​mspace width="​thinmathspace"/><​mi>​x</​mi><​mo>​=</​mo><​mi>​ln</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mrow><​mo>​(</​mo><​mrow><​mi>​x</​mi><​mo>​+</​mo><​mrow class="​MJX-TeXAtom-ORD"><​msqrt><​msup><​mi>​x</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup><​mo>​+</​mo><​mn>​1</​mn></​msqrt></​mrow></​mrow><​mo>​)</​mo></​mrow></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle operatorname {arsinh} ,x=ln left(x+{sqrt {x^{2}+1}}right)}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​b5e701ed143af79d5f221fcceeb826d2f5caf49c"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -1.838ex; width:​29.02ex;​ height:​4.843ex;"​ alt="​{displaystyle operatorname {arsinh} ,x=ln left(x+{sqrt {x^{2}+1}}right)}"/></​span></​dd></​dl><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle operatorname {arcosh} ,x=ln left(x+{sqrt {x^{2}-1}}right);​xgeq 1}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​arcosh</​mi><​mspace width="​thinmathspace"/><​mi>​x</​mi><​mo>​=</​mo><​mi>​ln</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mrow><​mo>​(</​mo><​mrow><​mi>​x</​mi><​mo>​+</​mo><​mrow class="​MJX-TeXAtom-ORD"><​msqrt><​msup><​mi>​x</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup><​mo>​−<​!-- &minus; --></​mo><​mn>​1</​mn></​msqrt></​mrow></​mrow><​mo>​)</​mo></​mrow><​mo>;</​mo><​mi>​x</​mi><​mo>​≥<​!-- &ge; --></​mo><​mn>​1</​mn></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle operatorname {arcosh} ,x=ln left(x+{sqrt {x^{2}-1}}right);​xgeq 1}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​1688b5a0f437d49ce9f92ab32cb57eb388c67d42"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -1.838ex; width:​35.9ex;​ height:​4.843ex;"​ alt="​{displaystyle operatorname {arcosh} ,x=ln left(x+{sqrt {x^{2}-1}}right);​xgeq 1}"/></​span></​dd></​dl><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle operatorname {artanh} ,x={tfrac {1}{2}}ln {frac {1+x}{1-x}};​left|xright|&​lt;​1}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​artanh</​mi><​mspace width="​thinmathspace"/><​mi>​x</​mi><​mo>​=</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​false"​ scriptlevel="​0"><​mfrac><​mn>​1</​mn><​mn>​2</​mn></​mfrac></​mstyle></​mrow><​mi>​ln</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mn>​1</​mn><​mo>​+</​mo><​mi>​x</​mi></​mrow><​mrow><​mn>​1</​mn><​mo>​−<​!-- &minus; --></​mo><​mi>​x</​mi></​mrow></​mfrac></​mrow><​mo>;</​mo><​mrow><​mo>​|</​mo><​mi>​x</​mi><​mo>​|</​mo></​mrow><​mo>&​lt;</​mo><​mn>​1</​mn></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle operatorname {artanh} ,x={tfrac {1}{2}}ln {frac {1+x}{1-x}};​left|xright|&​lt;​1}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​f410e5162aacf40024f94f492ec3f3bf827c8079"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -2.005ex; width:​30.387ex;​ height:​5.343ex;"​ alt="​{displaystyle operatorname {artanh} ,x={tfrac {1}{2}}ln {frac {1+x}{1-x}};​left|xright|&​lt;​1}"/></​span></​dd></​dl><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle operatorname {arsech} ,x=ln {frac {1+{sqrt {1-x^{2}}}}{x}};​0&​lt;​xleq 1}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​arsech</​mi><​mspace width="​thinmathspace"/><​mi>​x</​mi><​mo>​=</​mo><​mi>​ln</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mn>​1</​mn><​mo>​+</​mo><​mrow class="​MJX-TeXAtom-ORD"><​msqrt><​mn>​1</​mn><​mo>​−<​!-- &minus; --></​mo><​msup><​mi>​x</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup></​msqrt></​mrow></​mrow><​mi>​x</​mi></​mfrac></​mrow><​mo>;</​mo><​mn>​0</​mn><​mo>&​lt;</​mo><​mi>​x</​mi><​mo>​≤<​!-- &le; --></​mo><​mn>​1</​mn></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle operatorname {arsech} ,x=ln {frac {1+{sqrt {1-x^{2}}}}{x}};​0&​lt;​xleq 1}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​e351bf3a14f037eb9443e5d1c49be1c0195e3ed2"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -1.838ex; width:​38.311ex;​ height:​6.176ex;"​ alt="​{displaystyle operatorname {arsech} ,x=ln {frac {1+{sqrt {1-x^{2}}}}{x}};​0&​lt;​xleq 1}"/></​span></​dd></​dl><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle operatorname {arcsch} ,x=ln left({frac {1}{x}}+{frac {sqrt {1+x^{2}}}{left|xright|}}right)}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​arcsch</​mi><​mspace width="​thinmathspace"/><​mi>​x</​mi><​mo>​=</​mo><​mi>​ln</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mrow><​mo>​(</​mo><​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mn>​1</​mn><​mi>​x</​mi></​mfrac></​mrow><​mo>​+</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​msqrt><​mn>​1</​mn><​mo>​+</​mo><​msup><​mi>​x</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup></​msqrt><​mrow><​mo>​|</​mo><​mi>​x</​mi><​mo>​|</​mo></​mrow></​mfrac></​mrow></​mrow><​mo>​)</​mo></​mrow></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle operatorname {arcsch} ,x=ln left({frac {1}{x}}+{frac {sqrt {1+x^{2}}}{left|xright|}}right)}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​2a6f4912bfd4e48061aed7309769488e1bb4ce4f"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -3.171ex; width:​31.724ex;​ height:​7.509ex;"​ alt="​{displaystyle operatorname {arcsch} ,x=ln left({frac {1}{x}}+{frac {sqrt {1+x^{2}}}{left|xright|}}right)}"/></​span></​dd></​dl><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle operatorname {arcoth} ,x={tfrac {1}{2}}ln {frac {x+1}{x-1}};​left|xright|&​gt;​1}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​arcoth</​mi><​mspace width="​thinmathspace"/><​mi>​x</​mi><​mo>​=</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​false"​ scriptlevel="​0"><​mfrac><​mn>​1</​mn><​mn>​2</​mn></​mfrac></​mstyle></​mrow><​mi>​ln</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mi>​x</​mi><​mo>​+</​mo><​mn>​1</​mn></​mrow><​mrow><​mi>​x</​mi><​mo>​−<​!-- &minus; --></​mo><​mn>​1</​mn></​mrow></​mfrac></​mrow><​mo>;</​mo><​mrow><​mo>​|</​mo><​mi>​x</​mi><​mo>​|</​mo></​mrow><​mo>&​gt;</​mo><​mn>​1</​mn></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle operatorname {arcoth} ,x={tfrac {1}{2}}ln {frac {x+1}{x-1}};​left|xright|&​gt;​1}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​d79b2e4c2ad723bbbcf425f7faf6b1c9103d86ea"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -2.005ex; width:​30.127ex;​ height:​5.343ex;"​ alt="​{displaystyle operatorname {arcoth} ,x={tfrac {1}{2}}ln {frac {x+1}{x-1}};​left|xright|&​gt;​1}"/></​span></​dd></​dl>​
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle {frac {d}{dx}}sinh x=cosh x,​}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mi>​d</​mi><​mrow><​mi>​d</​mi><​mi>​x</​mi></​mrow></​mfrac></​mrow><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mo>​=</​mo><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mspace width="​thinmathspace"/></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle {frac {d}{dx}}sinh x=cosh x,​}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​4530dba92a47c444fb55137e41cde02f10800a30"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -2.005ex; width:​19.24ex;​ height:​5.509ex;"​ alt="​{displaystyle {frac {d}{dx}}sinh x=cosh x,​}"/></​span></​dd></​dl><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle {frac {d}{dx}}cosh x=sinh x,​}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mi>​d</​mi><​mrow><​mi>​d</​mi><​mi>​x</​mi></​mrow></​mfrac></​mrow><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mo>​=</​mo><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mspace width="​thinmathspace"/></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle {frac {d}{dx}}cosh x=sinh x,​}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​c050e85b31834b0df9a4a4ac863c715ea2e6ed7c"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -2.005ex; width:​19.24ex;​ height:​5.509ex;"​ alt="​{displaystyle {frac {d}{dx}}cosh x=sinh x,​}"/></​span></​dd></​dl><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle {frac {d}{dx}}tanh x=1-tanh ^{2}x={hbox{sech}}^{2}x=1/​cosh ^{2}x,​}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mi>​d</​mi><​mrow><​mi>​d</​mi><​mi>​x</​mi></​mrow></​mfrac></​mrow><​mi>​tanh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mo>​=</​mo><​mn>​1</​mn><​mo>​−<​!-- &minus; --></​mo><​msup><​mi>​tanh</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mo>​=</​mo><​msup><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​false"​ scriptlevel="​0"><​mtext>​sech</​mtext></​mstyle></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup><​mi>​x</​mi><​mo>​=</​mo><​mn>​1</​mn><​mrow class="​MJX-TeXAtom-ORD"><​mo>/</​mo></​mrow><​msup><​mi>​cosh</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mspace width="​thinmathspace"/></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle {frac {d}{dx}}tanh x=1-tanh ^{2}x={hbox{sech}}^{2}x=1/​cosh ^{2}x,​}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​48e115d0848c424686f85f8d33cfe9f81c95cd58"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -2.005ex; width:​47.79ex;​ height:​5.509ex;"​ alt="​{displaystyle {frac {d}{dx}}tanh x=1-tanh ^{2}x={hbox{sech}}^{2}x=1/​cosh ^{2}x,​}"/></​span></​dd></​dl><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle {frac {d}{dx}}coth x=1-coth ^{2}x=-{hbox{csch}}^{2}x=-1/​sinh ^{2}x,​}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mi>​d</​mi><​mrow><​mi>​d</​mi><​mi>​x</​mi></​mrow></​mfrac></​mrow><​mi>​coth</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mo>​=</​mo><​mn>​1</​mn><​mo>​−<​!-- &minus; --></​mo><​msup><​mi>​coth</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mo>​=</​mo><​mo>​−<​!-- &minus; --></​mo><​msup><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​false"​ scriptlevel="​0"><​mtext>​csch</​mtext></​mstyle></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup><​mi>​x</​mi><​mo>​=</​mo><​mo>​−<​!-- &minus; --></​mo><​mn>​1</​mn><​mrow class="​MJX-TeXAtom-ORD"><​mo>/</​mo></​mrow><​msup><​mi>​sinh</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mspace width="​thinmathspace"/></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle {frac {d}{dx}}coth x=1-coth ^{2}x=-{hbox{csch}}^{2}x=-1/​sinh ^{2}x,​}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​8e86b0ab330dedde33fa901922f6e848b765253f"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -2.005ex; width:​50.631ex;​ height:​5.509ex;"​ alt="​{displaystyle {frac {d}{dx}}coth x=1-coth ^{2}x=-{hbox{csch}}^{2}x=-1/​sinh ^{2}x,​}"/></​span></​dd></​dl><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle {frac {d}{dx}} {hbox{csch}},​x=-coth x {hbox{csch}},​x,​}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mi>​d</​mi><​mrow><​mi>​d</​mi><​mi>​x</​mi></​mrow></​mfrac></​mrow><​mtext>​ </​mtext><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​false"​ scriptlevel="​0"><​mtext>​csch</​mtext></​mstyle></​mrow><​mspace width="​thinmathspace"/><​mi>​x</​mi><​mo>​=</​mo><​mo>​−<​!-- &minus; --></​mo><​mi>​coth</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mtext>​ </​mtext><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​false"​ scriptlevel="​0"><​mtext>​csch</​mtext></​mstyle></​mrow><​mspace width="​thinmathspace"/><​mi>​x</​mi><​mspace width="​thinmathspace"/></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle {frac {d}{dx}} {hbox{csch}},​x=-coth x {hbox{csch}},​x,​}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​33cb5291f1e469a14a5ba134f3717201e578db4d"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -2.005ex; width:​28.313ex;​ height:​5.509ex;"​ alt="​{displaystyle {frac {d}{dx}} {hbox{csch}},​x=-coth x {hbox{csch}},​x,​}"/></​span></​dd></​dl><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle {frac {d}{dx}} {hbox{sech}},​x=-tanh x {hbox{sech}},​x,​}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mi>​d</​mi><​mrow><​mi>​d</​mi><​mi>​x</​mi></​mrow></​mfrac></​mrow><​mtext>​ </​mtext><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​false"​ scriptlevel="​0"><​mtext>​sech</​mtext></​mstyle></​mrow><​mspace width="​thinmathspace"/><​mi>​x</​mi><​mo>​=</​mo><​mo>​−<​!-- &minus; --></​mo><​mi>​tanh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mtext>​ </​mtext><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​false"​ scriptlevel="​0"><​mtext>​sech</​mtext></​mstyle></​mrow><​mspace width="​thinmathspace"/><​mi>​x</​mi><​mspace width="​thinmathspace"/></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle {frac {d}{dx}} {hbox{sech}},​x=-tanh x {hbox{sech}},​x,​}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​9b7b677943bf79f84918d5a5fecaefa4416c874d"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -2.005ex; width:​28.573ex;​ height:​5.509ex;"​ alt="​{displaystyle {frac {d}{dx}} {hbox{sech}},​x=-tanh x {hbox{sech}},​x,​}"/></​span></​dd></​dl><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle {frac {d}{dx}},​operatorname {arsinh} ,x={frac {1}{sqrt {x^{2}+1}}}}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mi>​d</​mi><​mrow><​mi>​d</​mi><​mi>​x</​mi></​mrow></​mfrac></​mrow><​mspace width="​thinmathspace"/><​mi>​arsinh</​mi><​mspace width="​thinmathspace"/><​mi>​x</​mi><​mo>​=</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mn>​1</​mn><​msqrt><​msup><​mi>​x</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup><​mo>​+</​mo><​mn>​1</​mn></​msqrt></​mfrac></​mrow></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle {frac {d}{dx}},​operatorname {arsinh} ,x={frac {1}{sqrt {x^{2}+1}}}}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​747a51097752199a518700dafdeab7a8c2e417e2"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -3.171ex; width:​25.127ex;​ height:​6.676ex;"​ alt="​{displaystyle {frac {d}{dx}},​operatorname {arsinh} ,x={frac {1}{sqrt {x^{2}+1}}}}"/></​span></​dd></​dl><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle {frac {d}{dx}},​operatorname {arcosh} ,x={frac {1}{sqrt {x^{2}-1}}}}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mi>​d</​mi><​mrow><​mi>​d</​mi><​mi>​x</​mi></​mrow></​mfrac></​mrow><​mspace width="​thinmathspace"/><​mi>​arcosh</​mi><​mspace width="​thinmathspace"/><​mi>​x</​mi><​mo>​=</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mn>​1</​mn><​msqrt><​msup><​mi>​x</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup><​mo>​−<​!-- &minus; --></​mo><​mn>​1</​mn></​msqrt></​mfrac></​mrow></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle {frac {d}{dx}},​operatorname {arcosh} ,x={frac {1}{sqrt {x^{2}-1}}}}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​e9239a0ffbfccf312d7d459682b528eaeac1300c"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -3.171ex; width:​25.383ex;​ height:​6.676ex;"​ alt="​{displaystyle {frac {d}{dx}},​operatorname {arcosh} ,x={frac {1}{sqrt {x^{2}-1}}}}"/></​span></​dd></​dl><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle {frac {d}{dx}},​operatorname {artanh} ,x={frac {1}{1-x^{2}}}}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mi>​d</​mi><​mrow><​mi>​d</​mi><​mi>​x</​mi></​mrow></​mfrac></​mrow><​mspace width="​thinmathspace"/><​mi>​artanh</​mi><​mspace width="​thinmathspace"/><​mi>​x</​mi><​mo>​=</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mn>​1</​mn><​mrow><​mn>​1</​mn><​mo>​−<​!-- &minus; --></​mo><​msup><​mi>​x</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup></​mrow></​mfrac></​mrow></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle {frac {d}{dx}},​operatorname {artanh} ,x={frac {1}{1-x^{2}}}}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​6c38462016475044f476f77c3f21b157e2c8c8ca"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -2.338ex; width:​23.307ex;​ height:​5.843ex;"​ alt="​{displaystyle {frac {d}{dx}},​operatorname {artanh} ,x={frac {1}{1-x^{2}}}}"/></​span></​dd></​dl><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle {frac {d}{dx}},​operatorname {arcsch} ,x=-{frac {1}{left|xright|{sqrt {1+x^{2}}}}}}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mi>​d</​mi><​mrow><​mi>​d</​mi><​mi>​x</​mi></​mrow></​mfrac></​mrow><​mspace width="​thinmathspace"/><​mi>​arcsch</​mi><​mspace width="​thinmathspace"/><​mi>​x</​mi><​mo>​=</​mo><​mo>​−<​!-- &minus; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mn>​1</​mn><​mrow><​mrow><​mo>​|</​mo><​mi>​x</​mi><​mo>​|</​mo></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​msqrt><​mn>​1</​mn><​mo>​+</​mo><​msup><​mi>​x</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup></​msqrt></​mrow></​mrow></​mfrac></​mrow></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle {frac {d}{dx}},​operatorname {arcsch} ,x=-{frac {1}{left|xright|{sqrt {1+x^{2}}}}}}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​9e54e187d635dc0bf0b42477b328d5914148fb89"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -3.171ex; width:​30.071ex;​ height:​6.676ex;"​ alt="​{displaystyle {frac {d}{dx}},​operatorname {arcsch} ,x=-{frac {1}{left|xright|{sqrt {1+x^{2}}}}}}"/></​span></​dd></​dl><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle {frac {d}{dx}},​operatorname {arsech} ,x=-{frac {1}{x{sqrt {1-x^{2}}}}}}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mi>​d</​mi><​mrow><​mi>​d</​mi><​mi>​x</​mi></​mrow></​mfrac></​mrow><​mspace width="​thinmathspace"/><​mi>​arsech</​mi><​mspace width="​thinmathspace"/><​mi>​x</​mi><​mo>​=</​mo><​mo>​−<​!-- &minus; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mn>​1</​mn><​mrow><​mi>​x</​mi><​mrow class="​MJX-TeXAtom-ORD"><​msqrt><​mn>​1</​mn><​mo>​−<​!-- &minus; --></​mo><​msup><​mi>​x</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup></​msqrt></​mrow></​mrow></​mfrac></​mrow></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle {frac {d}{dx}},​operatorname {arsech} ,x=-{frac {1}{x{sqrt {1-x^{2}}}}}}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​6ade8b93011edc46366d215c61decda1e3be6df6"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -3.171ex; width:​28.39ex;​ height:​6.676ex;"​ alt="​{displaystyle {frac {d}{dx}},​operatorname {arsech} ,x=-{frac {1}{x{sqrt {1-x^{2}}}}}}"/></​span></​dd></​dl><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle {frac {d}{dx}},​operatorname {arcoth} ,x={frac {1}{1-x^{2}}}}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mi>​d</​mi><​mrow><​mi>​d</​mi><​mi>​x</​mi></​mrow></​mfrac></​mrow><​mspace width="​thinmathspace"/><​mi>​arcoth</​mi><​mspace width="​thinmathspace"/><​mi>​x</​mi><​mo>​=</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mn>​1</​mn><​mrow><​mn>​1</​mn><​mo>​−<​!-- &minus; --></​mo><​msup><​mi>​x</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup></​mrow></​mfrac></​mrow></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle {frac {d}{dx}},​operatorname {arcoth} ,x={frac {1}{1-x^{2}}}}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​4bb652017248dd7b8b6fae8f551ab701eaf473ca"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -2.338ex; width:​23.047ex;​ height:​5.843ex;"​ alt="​{displaystyle {frac {d}{dx}},​operatorname {arcoth} ,x={frac {1}{1-x^{2}}}}"/></​span></​dd></​dl>​
 +<​p>​Xem thêm: <a href="​http://​vi.wikipedia.org/​w/​index.php?​title=Danh_s%C3%A1ch_t%C3%ADch_ph%C3%A2n_v%E1%BB%9Bi_h%C3%A0m_hyperbolic&​amp;​action=edit&​amp;​redlink=1"​ class="​new"​ title="​Danh sách tích phân với hàm hyperbolic (trang chưa được viết)">​Danh sách tích phân với hàm hyperbolic</​a>​
 +</p>
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle int sinh ax,​dx=a^{-1}cosh ax+C}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mo>​∫<​!-- &int; --></​mo><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​a</​mi><​mi>​x</​mi><​mspace width="​thinmathspace"/><​mi>​d</​mi><​mi>​x</​mi><​mo>​=</​mo><​msup><​mi>​a</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mo>​−<​!-- &minus; --></​mo><​mn>​1</​mn></​mrow></​msup><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​a</​mi><​mi>​x</​mi><​mo>​+</​mo><​mi>​C</​mi></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle int sinh ax,​dx=a^{-1}cosh ax+C}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​5e710f060e988701918a04c2b955b7f43b5cc0ee"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -2.338ex; width:​31.613ex;​ height:​5.676ex;"​ alt="​{displaystyle int sinh ax,​dx=a^{-1}cosh ax+C}"/></​span></​dd></​dl><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle int cosh ax,​dx=a^{-1}sinh ax+C}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mo>​∫<​!-- &int; --></​mo><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​a</​mi><​mi>​x</​mi><​mspace width="​thinmathspace"/><​mi>​d</​mi><​mi>​x</​mi><​mo>​=</​mo><​msup><​mi>​a</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mo>​−<​!-- &minus; --></​mo><​mn>​1</​mn></​mrow></​msup><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​a</​mi><​mi>​x</​mi><​mo>​+</​mo><​mi>​C</​mi></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle int cosh ax,​dx=a^{-1}sinh ax+C}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​3491402308815ddd5b154b69a2d75b133ef5a354"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -2.338ex; width:​31.613ex;​ height:​5.676ex;"​ alt="​{displaystyle int cosh ax,​dx=a^{-1}sinh ax+C}"/></​span></​dd></​dl><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle int tanh ax,​dx=a^{-1}ln(cosh ax)+C}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mo>​∫<​!-- &int; --></​mo><​mi>​tanh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​a</​mi><​mi>​x</​mi><​mspace width="​thinmathspace"/><​mi>​d</​mi><​mi>​x</​mi><​mo>​=</​mo><​msup><​mi>​a</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mo>​−<​!-- &minus; --></​mo><​mn>​1</​mn></​mrow></​msup><​mi>​ln</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​a</​mi><​mi>​x</​mi><​mo stretchy="​false">​)</​mo><​mo>​+</​mo><​mi>​C</​mi></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle int tanh ax,​dx=a^{-1}ln(cosh ax)+C}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​d93bda6875c7236075e7291e9a0812e060b7b807"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -2.338ex; width:​35.866ex;​ height:​5.676ex;"​ alt="​{displaystyle int tanh ax,​dx=a^{-1}ln(cosh ax)+C}"/></​span></​dd></​dl><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle int coth ax,​dx=a^{-1}ln(sinh ax)+C}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mo>​∫<​!-- &int; --></​mo><​mi>​coth</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​a</​mi><​mi>​x</​mi><​mspace width="​thinmathspace"/><​mi>​d</​mi><​mi>​x</​mi><​mo>​=</​mo><​msup><​mi>​a</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mo>​−<​!-- &minus; --></​mo><​mn>​1</​mn></​mrow></​msup><​mi>​ln</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​a</​mi><​mi>​x</​mi><​mo stretchy="​false">​)</​mo><​mo>​+</​mo><​mi>​C</​mi></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle int coth ax,​dx=a^{-1}ln(sinh ax)+C}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​6bc69d4568fc09aae97dc89a4bfca2af26bb2c94"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -2.338ex; width:​35.35ex;​ height:​5.676ex;"​ alt="​{displaystyle int coth ax,​dx=a^{-1}ln(sinh ax)+C}"/></​span></​dd></​dl><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle int {frac {du}{sqrt {a^{2}+u^{2}}}}=sinh ^{-1}left({frac {u}{a}}right)+C}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mo>​∫<​!-- &int; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mi>​d</​mi><​mi>​u</​mi></​mrow><​msqrt><​msup><​mi>​a</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup><​mo>​+</​mo><​msup><​mi>​u</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup></​msqrt></​mfrac></​mrow><​mo>​=</​mo><​msup><​mi>​sinh</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mo>​−<​!-- &minus; --></​mo><​mn>​1</​mn></​mrow></​msup><​mo>​⁡<​!-- &#8289; --></​mo><​mrow><​mo>​(</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mi>​u</​mi><​mi>​a</​mi></​mfrac></​mrow><​mo>​)</​mo></​mrow><​mo>​+</​mo><​mi>​C</​mi></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle int {frac {du}{sqrt {a^{2}+u^{2}}}}=sinh ^{-1}left({frac {u}{a}}right)+C}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​7a8a0b808aa718b091bbf9c44f790e7cbc804cfe"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -3.171ex; width:​32.376ex;​ height:​6.676ex;"​ alt="​{displaystyle int {frac {du}{sqrt {a^{2}+u^{2}}}}=sinh ^{-1}left({frac {u}{a}}right)+C}"/></​span></​dd></​dl><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle int {frac {du}{sqrt {u^{2}-a^{2}}}}=cosh ^{-1}left({frac {u}{a}}right)+C}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mo>​∫<​!-- &int; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mi>​d</​mi><​mi>​u</​mi></​mrow><​msqrt><​msup><​mi>​u</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup><​mo>​−<​!-- &minus; --></​mo><​msup><​mi>​a</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup></​msqrt></​mfrac></​mrow><​mo>​=</​mo><​msup><​mi>​cosh</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mo>​−<​!-- &minus; --></​mo><​mn>​1</​mn></​mrow></​msup><​mo>​⁡<​!-- &#8289; --></​mo><​mrow><​mo>​(</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mi>​u</​mi><​mi>​a</​mi></​mfrac></​mrow><​mo>​)</​mo></​mrow><​mo>​+</​mo><​mi>​C</​mi></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle int {frac {du}{sqrt {u^{2}-a^{2}}}}=cosh ^{-1}left({frac {u}{a}}right)+C}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​145ce4743614ca7fdb7a4d29b9eaabee336dd52d"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -3.171ex; width:​32.632ex;​ height:​6.676ex;"​ alt="​{displaystyle int {frac {du}{sqrt {u^{2}-a^{2}}}}=cosh ^{-1}left({frac {u}{a}}right)+C}"/></​span></​dd></​dl><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle int {frac {du}{a^{2}-u^{2}}}=a^{-1}tanh ^{-1}left({frac {u}{a}}right)+C;​u^{2}&​lt;​a^{2}}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mo>​∫<​!-- &int; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mi>​d</​mi><​mi>​u</​mi></​mrow><​mrow><​msup><​mi>​a</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup><​mo>​−<​!-- &minus; --></​mo><​msup><​mi>​u</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup></​mrow></​mfrac></​mrow><​mo>​=</​mo><​msup><​mi>​a</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mo>​−<​!-- &minus; --></​mo><​mn>​1</​mn></​mrow></​msup><​msup><​mi>​tanh</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mo>​−<​!-- &minus; --></​mo><​mn>​1</​mn></​mrow></​msup><​mo>​⁡<​!-- &#8289; --></​mo><​mrow><​mo>​(</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mi>​u</​mi><​mi>​a</​mi></​mfrac></​mrow><​mo>​)</​mo></​mrow><​mo>​+</​mo><​mi>​C</​mi><​mo>;</​mo><​msup><​mi>​u</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup><​mo>&​lt;</​mo><​msup><​mi>​a</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle int {frac {du}{a^{2}-u^{2}}}=a^{-1}tanh ^{-1}left({frac {u}{a}}right)+C;​u^{2}&​lt;​a^{2}}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​f16a1227484b01fcea84f6769b5d96fa8c721c6b"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -2.338ex; width:​43.307ex;​ height:​5.843ex;"​ alt="​{displaystyle int {frac {du}{a^{2}-u^{2}}}=a^{-1}tanh ^{-1}left({frac {u}{a}}right)+C;​u^{2}&​lt;​a^{2}}"/></​span></​dd></​dl><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle int {frac {du}{a^{2}-u^{2}}}=a^{-1}coth ^{-1}left({frac {u}{a}}right)+C;​u^{2}&​gt;​a^{2}}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mo>​∫<​!-- &int; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mi>​d</​mi><​mi>​u</​mi></​mrow><​mrow><​msup><​mi>​a</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup><​mo>​−<​!-- &minus; --></​mo><​msup><​mi>​u</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup></​mrow></​mfrac></​mrow><​mo>​=</​mo><​msup><​mi>​a</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mo>​−<​!-- &minus; --></​mo><​mn>​1</​mn></​mrow></​msup><​msup><​mi>​coth</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mo>​−<​!-- &minus; --></​mo><​mn>​1</​mn></​mrow></​msup><​mo>​⁡<​!-- &#8289; --></​mo><​mrow><​mo>​(</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mi>​u</​mi><​mi>​a</​mi></​mfrac></​mrow><​mo>​)</​mo></​mrow><​mo>​+</​mo><​mi>​C</​mi><​mo>;</​mo><​msup><​mi>​u</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup><​mo>&​gt;</​mo><​msup><​mi>​a</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle int {frac {du}{a^{2}-u^{2}}}=a^{-1}coth ^{-1}left({frac {u}{a}}right)+C;​u^{2}&​gt;​a^{2}}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​575a441e187d7a03aed2d75b2d9fde08ce33c97c"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -2.338ex; width:​43.047ex;​ height:​5.843ex;"​ alt="​{displaystyle int {frac {du}{a^{2}-u^{2}}}=a^{-1}coth ^{-1}left({frac {u}{a}}right)+C;​u^{2}&​gt;​a^{2}}"/></​span></​dd></​dl><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle int {frac {du}{u{sqrt {a^{2}-u^{2}}}}}=-a^{-1}operatorname {sech} ^{-1}left({frac {u}{a}}right)+C}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mo>​∫<​!-- &int; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mi>​d</​mi><​mi>​u</​mi></​mrow><​mrow><​mi>​u</​mi><​mrow class="​MJX-TeXAtom-ORD"><​msqrt><​msup><​mi>​a</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup><​mo>​−<​!-- &minus; --></​mo><​msup><​mi>​u</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup></​msqrt></​mrow></​mrow></​mfrac></​mrow><​mo>​=</​mo><​mo>​−<​!-- &minus; --></​mo><​msup><​mi>​a</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mo>​−<​!-- &minus; --></​mo><​mn>​1</​mn></​mrow></​msup><​msup><​mi>​sech</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mo>​−<​!-- &minus; --></​mo><​mn>​1</​mn></​mrow></​msup><​mo>​⁡<​!-- &#8289; --></​mo><​mrow><​mo>​(</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mi>​u</​mi><​mi>​a</​mi></​mfrac></​mrow><​mo>​)</​mo></​mrow><​mo>​+</​mo><​mi>​C</​mi></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle int {frac {du}{u{sqrt {a^{2}-u^{2}}}}}=-a^{-1}operatorname {sech} ^{-1}left({frac {u}{a}}right)+C}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​67e951e737420d3713e7ff7014532351e822b6fc"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -3.171ex; width:​39.589ex;​ height:​6.676ex;"​ alt="​{displaystyle int {frac {du}{u{sqrt {a^{2}-u^{2}}}}}=-a^{-1}operatorname {sech} ^{-1}left({frac {u}{a}}right)+C}"/></​span></​dd></​dl><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle int {frac {du}{u{sqrt {a^{2}+u^{2}}}}}=-a^{-1}operatorname {csch} ^{-1}left|{frac {u}{a}}right|+C}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mo>​∫<​!-- &int; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mi>​d</​mi><​mi>​u</​mi></​mrow><​mrow><​mi>​u</​mi><​mrow class="​MJX-TeXAtom-ORD"><​msqrt><​msup><​mi>​a</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup><​mo>​+</​mo><​msup><​mi>​u</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup></​msqrt></​mrow></​mrow></​mfrac></​mrow><​mo>​=</​mo><​mo>​−<​!-- &minus; --></​mo><​msup><​mi>​a</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mo>​−<​!-- &minus; --></​mo><​mn>​1</​mn></​mrow></​msup><​msup><​mi>​csch</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mo>​−<​!-- &minus; --></​mo><​mn>​1</​mn></​mrow></​msup><​mo>​⁡<​!-- &#8289; --></​mo><​mrow><​mo>​|</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mi>​u</​mi><​mi>​a</​mi></​mfrac></​mrow><​mo>​|</​mo></​mrow><​mo>​+</​mo><​mi>​C</​mi></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle int {frac {du}{u{sqrt {a^{2}+u^{2}}}}}=-a^{-1}operatorname {csch} ^{-1}left|{frac {u}{a}}right|+C}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​f93de57764e69926aaf091d0dd688b02a1f90e08"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -3.171ex; width:​38.108ex;​ height:​6.676ex;"​ alt="​{displaystyle int {frac {du}{u{sqrt {a^{2}+u^{2}}}}}=-a^{-1}operatorname {csch} ^{-1}left|{frac {u}{a}}right|+C}"/></​span></​dd></​dl><​p>​với <​i>​C</​i>​ là <a href="​http://​vi.wikipedia.org/​wiki/​H%E1%BA%B1ng_s%E1%BB%91_t%C3%ADch_ph%C3%A2n"​ title="​Hằng số tích phân">​hằng số tích phân</​a>​.
 +</p>
 +
 +<p>Ta có thể biểu diễn các hàm hyperbolic bằng <a href="​http://​vi.wikipedia.org/​wiki/​Chu%E1%BB%97i_Taylor"​ title="​Chuỗi Taylor">​chuỗi Taylor</​a>:​
 +</p>
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle sinh x=x+{frac {x^{3}}{3!}}+{frac {x^{5}}{5!}}+{frac {x^{7}}{7!}}+cdots =sum _{n=0}^{infty }{frac {x^{2n+1}}{(2n+1)!}}}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mo>​=</​mo><​mi>​x</​mi><​mo>​+</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​msup><​mi>​x</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​3</​mn></​mrow></​msup><​mrow><​mn>​3</​mn><​mo>​!</​mo></​mrow></​mfrac></​mrow><​mo>​+</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​msup><​mi>​x</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​5</​mn></​mrow></​msup><​mrow><​mn>​5</​mn><​mo>​!</​mo></​mrow></​mfrac></​mrow><​mo>​+</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​msup><​mi>​x</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​7</​mn></​mrow></​msup><​mrow><​mn>​7</​mn><​mo>​!</​mo></​mrow></​mfrac></​mrow><​mo>​+</​mo><​mo>​⋯<​!-- &#8943; --></​mo><​mo>​=</​mo><​munderover><​mo>​∑<​!-- &sum; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi>​n</​mi><​mo>​=</​mo><​mn>​0</​mn></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​normal">​∞<​!-- &infin; --></​mi></​mrow></​munderover><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​msup><​mi>​x</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn><​mi>​n</​mi><​mo>​+</​mo><​mn>​1</​mn></​mrow></​msup><​mrow><​mo stretchy="​false">​(</​mo><​mn>​2</​mn><​mi>​n</​mi><​mo>​+</​mo><​mn>​1</​mn><​mo stretchy="​false">​)</​mo><​mo>​!</​mo></​mrow></​mfrac></​mrow></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle sinh x=x+{frac {x^{3}}{3!}}+{frac {x^{5}}{5!}}+{frac {x^{7}}{7!}}+cdots =sum _{n=0}^{infty }{frac {x^{2n+1}}{(2n+1)!}}}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​5136eef875e847483a99cd2fdeb3fe99ed38ce76"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -3.005ex; width:​50.731ex;​ height:​6.843ex;"​ alt="​{displaystyle sinh x=x+{frac {x^{3}}{3!}}+{frac {x^{5}}{5!}}+{frac {x^{7}}{7!}}+cdots =sum _{n=0}^{infty }{frac {x^{2n+1}}{(2n+1)!}}}"/></​span></​dd></​dl><​p>​Hàm sinh <​i>​x</​i>​ biểu diễn theo chuỗi Taylor chỉ với số mũ lẻ của <​i>​x</​i>​. Do vậy nó là hàm lẻ, hay, −sinh <​i>​x</​i>​ = sinh(−<​i>​x</​i>​),​ và sinh 0 = 0.
 +</p>
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle cosh x=1+{frac {x^{2}}{2!}}+{frac {x^{4}}{4!}}+{frac {x^{6}}{6!}}+cdots =sum _{n=0}^{infty }{frac {x^{2n}}{(2n)!}}}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mo>​=</​mo><​mn>​1</​mn><​mo>​+</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​msup><​mi>​x</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup><​mrow><​mn>​2</​mn><​mo>​!</​mo></​mrow></​mfrac></​mrow><​mo>​+</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​msup><​mi>​x</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​4</​mn></​mrow></​msup><​mrow><​mn>​4</​mn><​mo>​!</​mo></​mrow></​mfrac></​mrow><​mo>​+</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​msup><​mi>​x</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​6</​mn></​mrow></​msup><​mrow><​mn>​6</​mn><​mo>​!</​mo></​mrow></​mfrac></​mrow><​mo>​+</​mo><​mo>​⋯<​!-- &#8943; --></​mo><​mo>​=</​mo><​munderover><​mo>​∑<​!-- &sum; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi>​n</​mi><​mo>​=</​mo><​mn>​0</​mn></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​normal">​∞<​!-- &infin; --></​mi></​mrow></​munderover><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​msup><​mi>​x</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn><​mi>​n</​mi></​mrow></​msup><​mrow><​mo stretchy="​false">​(</​mo><​mn>​2</​mn><​mi>​n</​mi><​mo stretchy="​false">​)</​mo><​mo>​!</​mo></​mrow></​mfrac></​mrow></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle cosh x=1+{frac {x^{2}}{2!}}+{frac {x^{4}}{4!}}+{frac {x^{6}}{6!}}+cdots =sum _{n=0}^{infty }{frac {x^{2n}}{(2n)!}}}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​ddc1170e1ca7c7a38152fcfe841b60deb418af4f"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -3.005ex; width:​46.816ex;​ height:​6.843ex;"​ alt="​{displaystyle cosh x=1+{frac {x^{2}}{2!}}+{frac {x^{4}}{4!}}+{frac {x^{6}}{6!}}+cdots =sum _{n=0}^{infty }{frac {x^{2n}}{(2n)!}}}"/></​span></​dd></​dl><​p>​Hàm cosh <​i>​x</​i>​ biểu diễn theo chuỗi Taylor chỉ với số mũ chẵn của <​i>​x</​i>​. Do vậy nó là <a href="​http://​vi.wikipedia.org/​w/​index.php?​title=H%C3%A0m_ch%E1%BA%B5n&​amp;​action=edit&​amp;​redlink=1"​ class="​new"​ title="​Hàm chẵn (trang chưa được viết)">​hàm chẵn</​a>,​ hay, nó đối xứng qua trục <​i>​y</​i>​. Tổng của chuỗi sinh và cosh là biểu thức <a href="​http://​vi.wikipedia.org/​wiki/​Chu%E1%BB%97i_v%C3%B4_h%E1%BA%A1n"​ class="​mw-redirect"​ title="​Chuỗi vô hạn">​chuỗi vô hạn</​a>​ của hàm mũ.
 +</p>
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle tanh x=x-{frac {x^{3}}{3}}+{frac {2x^{5}}{15}}-{frac {17x^{7}}{315}}+cdots =sum _{n=1}^{infty }{frac {2^{2n}(2^{2n}-1)B_{2n}x^{2n-1}}{(2n)!}},​left|xright|&​lt;​{frac {pi }{2}}}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​tanh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mo>​=</​mo><​mi>​x</​mi><​mo>​−<​!-- &minus; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​msup><​mi>​x</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​3</​mn></​mrow></​msup><​mn>​3</​mn></​mfrac></​mrow><​mo>​+</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mn>​2</​mn><​msup><​mi>​x</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​5</​mn></​mrow></​msup></​mrow><​mn>​15</​mn></​mfrac></​mrow><​mo>​−<​!-- &minus; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mn>​17</​mn><​msup><​mi>​x</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​7</​mn></​mrow></​msup></​mrow><​mn>​315</​mn></​mfrac></​mrow><​mo>​+</​mo><​mo>​⋯<​!-- &#8943; --></​mo><​mo>​=</​mo><​munderover><​mo>​∑<​!-- &sum; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi>​n</​mi><​mo>​=</​mo><​mn>​1</​mn></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​normal">​∞<​!-- &infin; --></​mi></​mrow></​munderover><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​msup><​mn>​2</​mn><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn><​mi>​n</​mi></​mrow></​msup><​mo stretchy="​false">​(</​mo><​msup><​mn>​2</​mn><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn><​mi>​n</​mi></​mrow></​msup><​mo>​−<​!-- &minus; --></​mo><​mn>​1</​mn><​mo stretchy="​false">​)</​mo><​msub><​mi>​B</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn><​mi>​n</​mi></​mrow></​msub><​msup><​mi>​x</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn><​mi>​n</​mi><​mo>​−<​!-- &minus; --></​mo><​mn>​1</​mn></​mrow></​msup></​mrow><​mrow><​mo stretchy="​false">​(</​mo><​mn>​2</​mn><​mi>​n</​mi><​mo stretchy="​false">​)</​mo><​mo>​!</​mo></​mrow></​mfrac></​mrow><​mo>,</​mo><​mrow><​mo>​|</​mo><​mi>​x</​mi><​mo>​|</​mo></​mrow><​mo>&​lt;</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mi>​π<​!-- &pi; --></​mi><​mn>​2</​mn></​mfrac></​mrow></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle tanh x=x-{frac {x^{3}}{3}}+{frac {2x^{5}}{15}}-{frac {17x^{7}}{315}}+cdots =sum _{n=1}^{infty }{frac {2^{2n}(2^{2n}-1)B_{2n}x^{2n-1}}{(2n)!}},​left|xright|&​lt;​{frac {pi }{2}}}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​c95c1e032cc52b10a5e058066523bcd4564f2143"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -3.005ex; width:​76.123ex;​ height:​7.009ex;"​ alt="​{displaystyle tanh x=x-{frac {x^{3}}{3}}+{frac {2x^{5}}{15}}-{frac {17x^{7}}{315}}+cdots =sum _{n=1}^{infty }{frac {2^{2n}(2^{2n}-1)B_{2n}x^{2n-1}}{(2n)!}},​left|xright|&​lt;​{frac {pi }{2}}}"/></​span></​dd></​dl><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle coth x=x^{-1}+{frac {x}{3}}-{frac {x^{3}}{45}}+{frac {2x^{5}}{945}}+cdots =x^{-1}+sum _{n=1}^{infty }{frac {2^{2n}B_{2n}x^{2n-1}}{(2n)!}},​0&​lt;​left|xright|&​lt;​pi }"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​coth</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mo>​=</​mo><​msup><​mi>​x</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mo>​−<​!-- &minus; --></​mo><​mn>​1</​mn></​mrow></​msup><​mo>​+</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mi>​x</​mi><​mn>​3</​mn></​mfrac></​mrow><​mo>​−<​!-- &minus; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​msup><​mi>​x</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​3</​mn></​mrow></​msup><​mn>​45</​mn></​mfrac></​mrow><​mo>​+</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mn>​2</​mn><​msup><​mi>​x</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​5</​mn></​mrow></​msup></​mrow><​mn>​945</​mn></​mfrac></​mrow><​mo>​+</​mo><​mo>​⋯<​!-- &#8943; --></​mo><​mo>​=</​mo><​msup><​mi>​x</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mo>​−<​!-- &minus; --></​mo><​mn>​1</​mn></​mrow></​msup><​mo>​+</​mo><​munderover><​mo>​∑<​!-- &sum; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi>​n</​mi><​mo>​=</​mo><​mn>​1</​mn></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​normal">​∞<​!-- &infin; --></​mi></​mrow></​munderover><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​msup><​mn>​2</​mn><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn><​mi>​n</​mi></​mrow></​msup><​msub><​mi>​B</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn><​mi>​n</​mi></​mrow></​msub><​msup><​mi>​x</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn><​mi>​n</​mi><​mo>​−<​!-- &minus; --></​mo><​mn>​1</​mn></​mrow></​msup></​mrow><​mrow><​mo stretchy="​false">​(</​mo><​mn>​2</​mn><​mi>​n</​mi><​mo stretchy="​false">​)</​mo><​mo>​!</​mo></​mrow></​mfrac></​mrow><​mo>,</​mo><​mn>​0</​mn><​mo>&​lt;</​mo><​mrow><​mo>​|</​mo><​mi>​x</​mi><​mo>​|</​mo></​mrow><​mo>&​lt;</​mo><​mi>​π<​!-- &pi; --></​mi></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle coth x=x^{-1}+{frac {x}{3}}-{frac {x^{3}}{45}}+{frac {2x^{5}}{945}}+cdots =x^{-1}+sum _{n=1}^{infty }{frac {2^{2n}B_{2n}x^{2n-1}}{(2n)!}},​0&​lt;​left|xright|&​lt;​pi }</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​832e455e65fcd25d3a9ff9dbb818754a029aaf59"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -3.005ex; width:​75.729ex;​ height:​6.843ex;"​ alt="​{displaystyle coth x=x^{-1}+{frac {x}{3}}-{frac {x^{3}}{45}}+{frac {2x^{5}}{945}}+cdots =x^{-1}+sum _{n=1}^{infty }{frac {2^{2n}B_{2n}x^{2n-1}}{(2n)!}},​0&​lt;​left|xright|&​lt;​pi }"/></​span>​ (<a href="​http://​vi.wikipedia.org/​wiki/​Chu%E1%BB%97i_Laurent"​ title="​Chuỗi Laurent">​chuỗi Laurent</​a>​)</​dd></​dl><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle operatorname {sech} ,x=1-{frac {x^{2}}{2}}+{frac {5x^{4}}{24}}-{frac {61x^{6}}{720}}+cdots =sum _{n=0}^{infty }{frac {E_{2n}x^{2n}}{(2n)!}},​left|xright|&​lt;​{frac {pi }{2}}}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​sech</​mi><​mspace width="​thinmathspace"/><​mi>​x</​mi><​mo>​=</​mo><​mn>​1</​mn><​mo>​−<​!-- &minus; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​msup><​mi>​x</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup><​mn>​2</​mn></​mfrac></​mrow><​mo>​+</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mn>​5</​mn><​msup><​mi>​x</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​4</​mn></​mrow></​msup></​mrow><​mn>​24</​mn></​mfrac></​mrow><​mo>​−<​!-- &minus; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mn>​61</​mn><​msup><​mi>​x</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​6</​mn></​mrow></​msup></​mrow><​mn>​720</​mn></​mfrac></​mrow><​mo>​+</​mo><​mo>​⋯<​!-- &#8943; --></​mo><​mo>​=</​mo><​munderover><​mo>​∑<​!-- &sum; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi>​n</​mi><​mo>​=</​mo><​mn>​0</​mn></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​normal">​∞<​!-- &infin; --></​mi></​mrow></​munderover><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​msub><​mi>​E</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn><​mi>​n</​mi></​mrow></​msub><​msup><​mi>​x</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn><​mi>​n</​mi></​mrow></​msup></​mrow><​mrow><​mo stretchy="​false">​(</​mo><​mn>​2</​mn><​mi>​n</​mi><​mo stretchy="​false">​)</​mo><​mo>​!</​mo></​mrow></​mfrac></​mrow><​mo>,</​mo><​mrow><​mo>​|</​mo><​mi>​x</​mi><​mo>​|</​mo></​mrow><​mo>&​lt;</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mi>​π<​!-- &pi; --></​mi><​mn>​2</​mn></​mfrac></​mrow></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle operatorname {sech} ,x=1-{frac {x^{2}}{2}}+{frac {5x^{4}}{24}}-{frac {61x^{6}}{720}}+cdots =sum _{n=0}^{infty }{frac {E_{2n}x^{2n}}{(2n)!}},​left|xright|&​lt;​{frac {pi }{2}}}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​8b380d5d7c7c0d493b34a9d5d38d9d6b123812a6"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -3.005ex; width:​61.597ex;​ height:​6.843ex;"​ alt="​{displaystyle operatorname {sech} ,x=1-{frac {x^{2}}{2}}+{frac {5x^{4}}{24}}-{frac {61x^{6}}{720}}+cdots =sum _{n=0}^{infty }{frac {E_{2n}x^{2n}}{(2n)!}},​left|xright|&​lt;​{frac {pi }{2}}}"/></​span></​dd></​dl><​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle operatorname {csch} ,​x=x^{-1}-{frac {x}{6}}+{frac {7x^{3}}{360}}-{frac {31x^{5}}{15120}}+cdots =x^{-1}+sum _{n=1}^{infty }{frac {2(1-2^{2n-1})B_{2n}x^{2n-1}}{(2n)!}},​0&​lt;​left|xright|&​lt;​pi }"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​csch</​mi><​mspace width="​thinmathspace"/><​mi>​x</​mi><​mo>​=</​mo><​msup><​mi>​x</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mo>​−<​!-- &minus; --></​mo><​mn>​1</​mn></​mrow></​msup><​mo>​−<​!-- &minus; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mi>​x</​mi><​mn>​6</​mn></​mfrac></​mrow><​mo>​+</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mn>​7</​mn><​msup><​mi>​x</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​3</​mn></​mrow></​msup></​mrow><​mn>​360</​mn></​mfrac></​mrow><​mo>​−<​!-- &minus; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mn>​31</​mn><​msup><​mi>​x</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​5</​mn></​mrow></​msup></​mrow><​mn>​15120</​mn></​mfrac></​mrow><​mo>​+</​mo><​mo>​⋯<​!-- &#8943; --></​mo><​mo>​=</​mo><​msup><​mi>​x</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mo>​−<​!-- &minus; --></​mo><​mn>​1</​mn></​mrow></​msup><​mo>​+</​mo><​munderover><​mo>​∑<​!-- &sum; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi>​n</​mi><​mo>​=</​mo><​mn>​1</​mn></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​normal">​∞<​!-- &infin; --></​mi></​mrow></​munderover><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mn>​2</​mn><​mo stretchy="​false">​(</​mo><​mn>​1</​mn><​mo>​−<​!-- &minus; --></​mo><​msup><​mn>​2</​mn><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn><​mi>​n</​mi><​mo>​−<​!-- &minus; --></​mo><​mn>​1</​mn></​mrow></​msup><​mo stretchy="​false">​)</​mo><​msub><​mi>​B</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn><​mi>​n</​mi></​mrow></​msub><​msup><​mi>​x</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn><​mi>​n</​mi><​mo>​−<​!-- &minus; --></​mo><​mn>​1</​mn></​mrow></​msup></​mrow><​mrow><​mo stretchy="​false">​(</​mo><​mn>​2</​mn><​mi>​n</​mi><​mo stretchy="​false">​)</​mo><​mo>​!</​mo></​mrow></​mfrac></​mrow><​mo>,</​mo><​mn>​0</​mn><​mo>&​lt;</​mo><​mrow><​mo>​|</​mo><​mi>​x</​mi><​mo>​|</​mo></​mrow><​mo>&​lt;</​mo><​mi>​π<​!-- &pi; --></​mi></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle operatorname {csch} ,​x=x^{-1}-{frac {x}{6}}+{frac {7x^{3}}{360}}-{frac {31x^{5}}{15120}}+cdots =x^{-1}+sum _{n=1}^{infty }{frac {2(1-2^{2n-1})B_{2n}x^{2n-1}}{(2n)!}},​0&​lt;​left|xright|&​lt;​pi }</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​b1c609320eb5ca753065d9840ec8c5a3dddf2ac3"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -3.005ex; width:​88.501ex;​ height:​7.009ex;"​ alt="​{displaystyle operatorname {csch} ,​x=x^{-1}-{frac {x}{6}}+{frac {7x^{3}}{360}}-{frac {31x^{5}}{15120}}+cdots =x^{-1}+sum _{n=1}^{infty }{frac {2(1-2^{2n-1})B_{2n}x^{2n-1}}{(2n)!}},​0&​lt;​left|xright|&​lt;​pi }"/></​span>​ (<a href="​http://​vi.wikipedia.org/​wiki/​Chu%E1%BB%97i_Laurent"​ title="​Chuỗi Laurent">​chuỗi Laurent</​a>​)</​dd></​dl><​p>​với
 +</p>
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle B_{n},​}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​msub><​mi>​B</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​n</​mi></​mrow></​msub><​mspace width="​thinmathspace"/></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle B_{n},​}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​d352186a495a156ca173e351226973b84706a165"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -0.671ex; width:​3.37ex;​ height:​2.509ex;"​ alt="​{displaystyle B_{n},​}"/></​span>​ là <a href="​http://​vi.wikipedia.org/​w/​index.php?​title=S%E1%BB%91_Bernoulli&​amp;​action=edit&​amp;​redlink=1"​ class="​new"​ title="​Số Bernoulli (trang chưa được viết)">​số Bernoulli</​a>​ thứ n</​dd>​
 +<​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle E_{n},​}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​msub><​mi>​E</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​n</​mi></​mrow></​msub><​mspace width="​thinmathspace"/></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle E_{n},​}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​5c621cabe4418802f7f26e069a046cd2270bff41"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -0.671ex; width:​3.321ex;​ height:​2.509ex;"​ alt="​{displaystyle E_{n},​}"/></​span>​ là <a href="​http://​vi.wikipedia.org/​wiki/​S%E1%BB%91_e"​ title="​Số e">​số Euler</​a>​ thứ n</​dd></​dl>​
 +<​p>​Từ định nghĩa của sinh và cosh hypebolic, ta có các đồng nhất thức sau:
 +</p>
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle e^{x}=cosh x+sinh x}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​msup><​mi>​e</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​x</​mi></​mrow></​msup><​mo>​=</​mo><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mo>​+</​mo><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle e^{x}=cosh x+sinh x}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​b6cbeacb0a2a95dc7efaa41963fab7b7d9efefe6"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -0.505ex; width:​20.18ex;​ height:​2.509ex;"​ alt="​{displaystyle e^{x}=cosh x+sinh x}"/></​span></​dd></​dl><​p>​và
 +</p>
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle e^{-x}=cosh x-sinh x}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​msup><​mi>​e</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mo>​−<​!-- &minus; --></​mo><​mi>​x</​mi></​mrow></​msup><​mo>​=</​mo><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mo>​−<​!-- &minus; --></​mo><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle e^{-x}=cosh x-sinh x}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​b3bc1cd4530b7a249e364da8f1acdc53fa07770f"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -0.505ex; width:​21.459ex;​ height:​2.676ex;"​ alt="​{displaystyle e^{-x}=cosh x-sinh x}"/></​span></​dd></​dl><​p>​Các biểu thức trên tương tự như các hàm sin và cosin, dựa trên công thức Euler, như là tổng của hai mũ lũy thừa.
 +</​p><​p>​Thêm vào đó,
 +</p>
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle e^{x}={sqrt {frac {1+tanh x}{1-tanh x}}}={frac {1+tanh {frac {x}{2}}}{1-tanh {frac {x}{2}}}}}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​msup><​mi>​e</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​x</​mi></​mrow></​msup><​mo>​=</​mo><​mrow class="​MJX-TeXAtom-ORD"><​msqrt><​mfrac><​mrow><​mn>​1</​mn><​mo>​+</​mo><​mi>​tanh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi></​mrow><​mrow><​mn>​1</​mn><​mo>​−<​!-- &minus; --></​mo><​mi>​tanh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi></​mrow></​mfrac></​msqrt></​mrow><​mo>​=</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mn>​1</​mn><​mo>​+</​mo><​mi>​tanh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mi>​x</​mi><​mn>​2</​mn></​mfrac></​mrow></​mrow><​mrow><​mn>​1</​mn><​mo>​−<​!-- &minus; --></​mo><​mi>​tanh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mi>​x</​mi><​mn>​2</​mn></​mfrac></​mrow></​mrow></​mfrac></​mrow></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle e^{x}={sqrt {frac {1+tanh x}{1-tanh x}}}={frac {1+tanh {frac {x}{2}}}{1-tanh {frac {x}{2}}}}}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​fa0159b0bba9bc859b2ba0e62083b5783a6a17b4"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -3.005ex; width:​33.639ex;​ height:​7.176ex;"​ alt="​{displaystyle e^{x}={sqrt {frac {1+tanh x}{1-tanh x}}}={frac {1+tanh {frac {x}{2}}}{1-tanh {frac {x}{2}}}}}"/></​span></​dd></​dl>​
 +<​p>​Vì hàm mũ được định nghĩa cho cả <a href="​http://​vi.wikipedia.org/​wiki/​S%E1%BB%91_ph%E1%BB%A9c"​ title="​Số phức">​số phức</​a>,​ có thể mở rộng định nghĩa hàm hypebolic cho các đối số phức. Khi ấy các hàm sinh <​i>​z</​i>​ và cosh <​i>​z</​i>​ là những <a href="​http://​vi.wikipedia.org/​w/​index.php?​title=H%C3%A0m_ch%E1%BB%89nh_h%C3%ACnh&​amp;​action=edit&​amp;​redlink=1"​ class="​new"​ title="​Hàm chỉnh hình (trang chưa được viết)">​hàm chỉnh hình</​a>​ (Holomorphic function).
 +</​p><​p>​Các mối liên hệ giữa các hàm lượng giác thường được cho bởi công thức Euler và áp dụng cho các biến phức:
 +</p>
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle {begin{aligned}e^{ix}&​amp;​=cos x+i;sin x\e^{-ix}&​amp;​=cos x-i;sin xend{aligned}}}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mrow class="​MJX-TeXAtom-ORD"><​mtable columnalign="​right left right left right left right left right left right left" rowspacing="​3pt"​ columnspacing="​0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="​true"><​mtr><​mtd><​msup><​mi>​e</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​i</​mi><​mi>​x</​mi></​mrow></​msup></​mtd><​mtd><​mi/><​mo>​=</​mo><​mi>​cos</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mo>​+</​mo><​mi>​i</​mi><​mspace width="​thickmathspace"/><​mi>​sin</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi></​mtd></​mtr><​mtr><​mtd><​msup><​mi>​e</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mo>​−<​!-- &minus; --></​mo><​mi>​i</​mi><​mi>​x</​mi></​mrow></​msup></​mtd><​mtd><​mi/><​mo>​=</​mo><​mi>​cos</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mo>​−<​!-- &minus; --></​mo><​mi>​i</​mi><​mspace width="​thickmathspace"/><​mi>​sin</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi></​mtd></​mtr></​mtable></​mrow></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle {begin{aligned}e^{ix}&​amp;​=cos x+i;sin x\e^{-ix}&​amp;​=cos x-i;sin xend{aligned}}}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​1cae5cad4d905fcbfe8dfb68f5d60c9da1663664"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -2.505ex; width:​22.027ex;​ height:​6.176ex;"​ alt="​{displaystyle {begin{aligned}e^{ix}&​amp;​=cos x+i;sin x\e^{-ix}&​amp;​=cos x-i;sin xend{aligned}}}"/></​span></​dd></​dl><​p>​do đó:
 +</p>
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle {begin{aligned}cosh(ix)&​amp;​={frac {1}{2}}left(e^{ix}+e^{-ix}right)=cos x\sinh(ix)&​amp;​={frac {1}{2}}left(e^{ix}-e^{-ix}right)=isin x\cosh(x+iy)&​amp;​=cosh(x)cos(y)+isinh(x)sin(y)\sinh(x+iy)&​amp;​=sinh(x)cos(y)+icosh(x)sin(y)\tanh(ix)&​amp;​=itan x\cosh x&​amp;​=cos(ix)\sinh x&​amp;​=-isin(ix)\tanh x&​amp;​=-itan(ix)end{aligned}}}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mrow class="​MJX-TeXAtom-ORD"><​mtable columnalign="​right left right left right left right left right left right left" rowspacing="​3pt"​ columnspacing="​0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="​true"><​mtr><​mtd><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​i</​mi><​mi>​x</​mi><​mo stretchy="​false">​)</​mo></​mtd><​mtd><​mi/><​mo>​=</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mn>​1</​mn><​mn>​2</​mn></​mfrac></​mrow><​mrow><​mo>​(</​mo><​mrow><​msup><​mi>​e</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​i</​mi><​mi>​x</​mi></​mrow></​msup><​mo>​+</​mo><​msup><​mi>​e</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mo>​−<​!-- &minus; --></​mo><​mi>​i</​mi><​mi>​x</​mi></​mrow></​msup></​mrow><​mo>​)</​mo></​mrow><​mo>​=</​mo><​mi>​cos</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi></​mtd></​mtr><​mtr><​mtd><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​i</​mi><​mi>​x</​mi><​mo stretchy="​false">​)</​mo></​mtd><​mtd><​mi/><​mo>​=</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mn>​1</​mn><​mn>​2</​mn></​mfrac></​mrow><​mrow><​mo>​(</​mo><​mrow><​msup><​mi>​e</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi>​i</​mi><​mi>​x</​mi></​mrow></​msup><​mo>​−<​!-- &minus; --></​mo><​msup><​mi>​e</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mo>​−<​!-- &minus; --></​mo><​mi>​i</​mi><​mi>​x</​mi></​mrow></​msup></​mrow><​mo>​)</​mo></​mrow><​mo>​=</​mo><​mi>​i</​mi><​mi>​sin</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi></​mtd></​mtr><​mtr><​mtd><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​x</​mi><​mo>​+</​mo><​mi>​i</​mi><​mi>​y</​mi><​mo stretchy="​false">​)</​mo></​mtd><​mtd><​mi/><​mo>​=</​mo><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​x</​mi><​mo stretchy="​false">​)</​mo><​mi>​cos</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​y</​mi><​mo stretchy="​false">​)</​mo><​mo>​+</​mo><​mi>​i</​mi><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​x</​mi><​mo stretchy="​false">​)</​mo><​mi>​sin</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​y</​mi><​mo stretchy="​false">​)</​mo></​mtd></​mtr><​mtr><​mtd><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​x</​mi><​mo>​+</​mo><​mi>​i</​mi><​mi>​y</​mi><​mo stretchy="​false">​)</​mo></​mtd><​mtd><​mi/><​mo>​=</​mo><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​x</​mi><​mo stretchy="​false">​)</​mo><​mi>​cos</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​y</​mi><​mo stretchy="​false">​)</​mo><​mo>​+</​mo><​mi>​i</​mi><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​x</​mi><​mo stretchy="​false">​)</​mo><​mi>​sin</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​y</​mi><​mo stretchy="​false">​)</​mo></​mtd></​mtr><​mtr><​mtd><​mi>​tanh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​i</​mi><​mi>​x</​mi><​mo stretchy="​false">​)</​mo></​mtd><​mtd><​mi/><​mo>​=</​mo><​mi>​i</​mi><​mi>​tan</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi></​mtd></​mtr><​mtr><​mtd><​mi>​cosh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi></​mtd><​mtd><​mi/><​mo>​=</​mo><​mi>​cos</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​i</​mi><​mi>​x</​mi><​mo stretchy="​false">​)</​mo></​mtd></​mtr><​mtr><​mtd><​mi>​sinh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi></​mtd><​mtd><​mi/><​mo>​=</​mo><​mo>​−<​!-- &minus; --></​mo><​mi>​i</​mi><​mi>​sin</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​i</​mi><​mi>​x</​mi><​mo stretchy="​false">​)</​mo></​mtd></​mtr><​mtr><​mtd><​mi>​tanh</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi></​mtd><​mtd><​mi/><​mo>​=</​mo><​mo>​−<​!-- &minus; --></​mo><​mi>​i</​mi><​mi>​tan</​mi><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​i</​mi><​mi>​x</​mi><​mo stretchy="​false">​)</​mo></​mtd></​mtr></​mtable></​mrow></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle {begin{aligned}cosh(ix)&​amp;​={frac {1}{2}}left(e^{ix}+e^{-ix}right)=cos x\sinh(ix)&​amp;​={frac {1}{2}}left(e^{ix}-e^{-ix}right)=isin x\cosh(x+iy)&​amp;​=cosh(x)cos(y)+isinh(x)sin(y)\sinh(x+iy)&​amp;​=sinh(x)cos(y)+icosh(x)sin(y)\tanh(ix)&​amp;​=itan x\cosh x&​amp;​=cos(ix)\sinh x&​amp;​=-isin(ix)\tanh x&​amp;​=-itan(ix)end{aligned}}}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​045e22a08388fdf8f17247fc365b3036179e73e3"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -14.171ex; width:​47.721ex;​ height:​29.509ex;"​ alt="​{displaystyle {begin{aligned}cosh(ix)&​amp;​={frac {1}{2}}left(e^{ix}+e^{-ix}right)=cos x\sinh(ix)&​amp;​={frac {1}{2}}left(e^{ix}-e^{-ix}right)=isin x\cosh(x+iy)&​amp;​=cosh(x)cos(y)+isinh(x)sin(y)\sinh(x+iy)&​amp;​=sinh(x)cos(y)+icosh(x)sin(y)\tanh(ix)&​amp;​=itan x\cosh x&​amp;​=cos(ix)\sinh x&​amp;​=-isin(ix)\tanh x&​amp;​=-itan(ix)end{aligned}}}"/></​span></​dd></​dl><​p>​Vì vậy các hàm hypebolic phức là những <a href="​http://​vi.wikipedia.org/​wiki/​H%C3%A0m_tu%E1%BA%A7n_ho%C3%A0n"​ title="​Hàm tuần hoàn">​hàm tuần hoàn</​a>​ theo phần ảo, với chu kỳ <span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle 2pi i}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mn>​2</​mn><​mi>​π<​!-- &pi; --></​mi><​mi>​i</​mi></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle 2pi i}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​7f5715af49984c5b33961d55f532d14497b0cbae"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -0.338ex; width:​3.297ex;​ height:​2.176ex;"​ alt="​{displaystyle 2pi i}"/></​span>​ (và <span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle pi i}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​π<​!-- &pi; --></​mi><​mi>​i</​mi></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle pi i}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​1f9b8c446f47536ec850ff65759e419f99e051ef"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -0.338ex; width:​2.134ex;​ height:​2.176ex;"​ alt="​{displaystyle pi i}"/></​span>​ cho các hàm tang và cotang hypebolic).
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10404-h-m-hypebolic-la-gi.txt · Last modified: 2018/11/07 17:08 (external edit)