雙曲扇形a的很多雙曲函数可以在几何上依据以O为中心的雙曲線来构造。
在数学中,雙曲函數恆等式是对出现的变量的所有值都为實的涉及到雙曲函數的等式。这些恒等式在表达式中有些雙曲函數需要简化的时候是很有用的。雙曲函數的恆等式有的與三角恆等式類似。就如同三角函數,他有一个重要应用是非雙曲函數的积分:一个常用技巧是首先使用换元积分法,規則與使用三角函数的代换规则類似,则通过雙曲函數恆等式可简化结果的积分。
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函数
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倒數函数
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全寫
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簡寫
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全寫
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簡寫
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函数
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hyperbolic sine
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sinh
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hyperbolic cosecant
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csch
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反函数
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inverse hyperbolic sine
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arcsinh
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inverse hyperbolic cosecant
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arccsch
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函数
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hyperbolic cosine
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cosh
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hyperbolic secant
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sech
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反函数
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inverse hyperbolic cosine
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arccosh
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inverse hyperbolic secant
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arcsech
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函数
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hyperbolic tangent
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tanh
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hyperbolic cotangent
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coth
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反函数
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inverse hyperbolic tangent
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arctanh
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inverse hyperbolic cotangent
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arccoth
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基本關係[编辑]
sinh, cosh 和 tanh
csch, sech 和 coth
雙曲函數基本恒等式如下:
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![{\displaystyle \sinh x={{e^{x}-e^{-x}} \over 2}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AatK0ate0aqhBoAhFztK2yjo5nDs0ajJAz2o0oDhEngo3nqi1nAnB)
![{\displaystyle \cosh x={{e^{x}+e^{-x}} \over 2}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84ajw3oNlCnjmPnqzCajsQnji3oDG4a2ePaNJCythEoNlEztBAzAo0)
![{\displaystyle \tanh x={{\sinh x} \over {\cosh x}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82zDoNzDaNaAs3ythCoNvCztG3nte3ygsPyjs3aAnDoArFotK3nDwN)
![{\displaystyle \coth x={1 \over {\tanh x}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CzqaOztBFoArEyjzByjhAzjhEyqw2oDa3zDJCago5yjhByjaQntFF)
![{\displaystyle {\mathop {\rm {sech}} }x={1 \over {\cosh x}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CnjFAygwOajzEaqsNyjnCoqvEajG3zNG3zja2yqw0zDlAngnCz2s4)
![{\displaystyle {\mathop {\rm {csch}} }x={1 \over {\sinh x}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QnjKQntC4nDC4a2iPz2eQyteQytCPaDoOzAzDzNnAzNs5zDe0yje5)
就如同三角函數,由上面的平方關係加上雙曲函數的基本定義,可以導出下面的表格,即每個雙曲函數都可以用其他五個表達。(严谨地说,所有根号前都应根据实际情况添加正负号)
函數
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sinh
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cosh
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tanh
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coth
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sech
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csch
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其他函數的基本關係[编辑]
三角函數還有正矢、餘矢、半正矢、半餘矢、外正割、外餘割等函數,利用他們的定義也可以導出雙曲函數。
名稱
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函數
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值
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雙曲正矢, hyperbolic versine
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![{\displaystyle \operatorname {versinh} (x)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NzNCOzqo3zjC1ntrCatvEyqdEaAi4ygaNntBEoNdFa2s0zAwNnjaP)
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雙曲餘矢, hyperbolic coversine
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![{\displaystyle \operatorname {coversinh} (x)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QzghFztG1ntJFz2aNzto0aqhAoAo0ngrAngvAntvAzAaOyjGQoDG1)
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雙曲半正矢 , hyperbolic haversine
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雙曲半餘矢 , hyperbolic hacoversine
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雙曲外正割 , hyperbolic exsecant
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雙曲外餘割 , hyperbolic excosecant
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和角公式[编辑]
![{\displaystyle \sinh(x+y)\ =\sinh x\cosh y+\cosh x\sinh y\,}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Fo2rDngw5oDaQzjKQaAhEzDm3ytrEotK0zDnBoDK0oNeOnjC1yjo1)
![{\displaystyle \sinh(x-y)\ =\sinh x\cosh y-\cosh x\sinh y\,}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80ajs1z2a2nqoNytsQnjs1zqw3aqs5a2sQoti4zNrDztdEnAe5yqo5)
![{\displaystyle \cosh(x+y)\ =\cosh x\cosh y+\sinh x\sinh y\,}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QnAiNzNw1nja1ntiNa2rFyje4ygzCa2nCntmOaDJBaDmOnjzCaDlA)
![{\displaystyle \cosh(x-y)\ =\cosh x\cosh y-\sinh x\sinh y\,}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QnqdDoAvBoNs0yjzBnqeQotlBnjsNotG1ajoNzjo2aNiNoNBAnAnD)
![{\displaystyle \tanh(x+y)\ ={\frac {\tanh x+\tanh y}{1+\tanh x\tanh y}}\,}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AzAnAnqe2z2a2zqnBngeNoAwPa2rEztC0zga5oNi3z2e3ajGQaAvB)
![{\displaystyle \tanh(x-y)\ ={\frac {\tanh x-\tanh y}{1-\tanh x\tanh y}}\,}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80aAaNo2zAz2s3oqeNnjsNyjoOzDvFnDo1zNa0yjzCotw0ngdCnjzA)
和差化積公式[编辑]
![{\displaystyle \sinh x+\sinh y\ =2\sinh {\frac {x+y}{2}}\cosh {\frac {x-y}{2}}\,}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81zjJByjm3zjBEzDw4ntm2ztsOoDmOntFDatrAyga5oqeOajw1ygwQ)
![{\displaystyle \sinh x-\sinh y\ =2\cosh {\frac {x+y}{2}}\sinh {\frac {x-y}{2}}\,}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DztwNyge4ztw0aqe2aqnFyghAntC3yqnEygaNzta3nDsPnDC0aqdA)
![{\displaystyle \cosh x+\cosh y\ =2\cosh {\frac {x+y}{2}}\cosh {\frac {x-y}{2}}\,}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QnqvBnDoPajmQnAw1a2a5zNsQzqzBajJAnDdCoNwQngw5aNa1njeN)
![{\displaystyle \cosh x-\cosh y\ =2\sinh {\frac {x+y}{2}}\sinh {\frac {x-y}{2}}\,}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DaDwPygrEaqdEaDK0aqwPo2a1aAvDajdAaDoOzqiOoAiOo2hCyjK0)
![{\displaystyle \tanh x+\tanh y\ ={\frac {\sinh(x+y)}{\cosh x\cosh y}}\,}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9EaDe2oAa1nDoOaAoOngvCoDG1othAajlBnDm3nDrEnAoOyghDatmP)
![{\displaystyle \tanh x-\tanh y\ ={\frac {\sinh(x-y)}{\cosh x\cosh y}}\,}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Eyqa4nDhAatzEzto2aDa3zqw2ntG5ztGQaqhBytK4ygiOaDG5yqw0)
積化和差公式[编辑]
![{\displaystyle \sinh x\sinh y\ ={\frac {\cosh(x+y)-\cosh(x-y)}{2}}\,}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NzDe0oqoOa2o4o2wNyqnBnjC4zAiPzNFAa2zEaAsPaqe2oDhDygrD)
![{\displaystyle \cosh x\cosh y\ ={\frac {\cosh(x+y)+\cosh(x-y)}{2}}\,}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9BaNGOnDBFygrDzta1oAo1nghEz2a0a2oOnDm0atiOytdAoNmNngw3)
![{\displaystyle \sinh x\cosh y\ ={\frac {\sinh(x+y)+\sinh(x-y)}{2}}\,}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83oqvFzqo2oAsNate5oqs0nji2nDG0aDsPztFBaAi4ntdFati0oArE)
倍角公式[编辑]
![{\displaystyle \sinh 2x\ =2\sinh x\cosh x\,}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Ba2zCytzCyqa0agdEags2ntnDnqsPaDs4nAzDztw2zqo2a2nCzjhA)
![{\displaystyle \cosh 2x\ =\cosh ^{2}x+\sinh ^{2}x=2\cosh ^{2}x-1=2\sinh ^{2}x+1\,}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81zgrEztG4ztnFytsQoAa2age2njsOzNrCnAa0ntvFnDiNagnDzNnD)
![{\displaystyle \tanh 2x\ ={\frac {2\tanh x}{1+\tanh ^{2}x}}\,}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DnArAoDi1nqiPaqnEaNFDyjBFzjhAnjdAntvAzNlEzNJDyqhDzNsN)
![{\displaystyle \sinh 3x\ =3\sinh x+4\sinh ^{3}x}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Nyta5ota5ats2zjGQnje4atrCoAzFzAaPztnAytBBotJBnDnAoDzE)
![{\displaystyle \cosh 3x\ =4\cosh ^{3}x-3\cosh x}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9BajBBaDa4nqnEygsQots4oDs3agw1ntvEzAePzgvBaghCzjBEo2e2)
半形公式[编辑]
![{\displaystyle \sinh {\frac {x}{2}}\ =\operatorname {sgn} x{\sqrt {\frac {\cosh x-1}{2}}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QnDFBoNKOajK0zDi2ntsOnAvAoNG0ajw1yqs3z2oNzNlDnjJAytdC)
![{\displaystyle \cosh {\frac {x}{2}}\ ={\sqrt {\frac {\cosh x+1}{2}}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OnjhCoNw3nDe0yqw2nji0atlBzAw4otvDa2e0zto0zgsQnDG0nqa5)
![{\displaystyle \tanh {\frac {x}{2}}\ ={\frac {\cosh x-1}{\sinh x}}\ ={\frac {\sinh x}{1+\cosh x}}\,}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81oNs2ngs2age4a2zCntdAaArEztmNntsQyqe3nAw3a2e1ztsOnthD)
幂简约公式[编辑]
![{\displaystyle \sinh ^{2}x={\frac {\cosh 2x-1}{2}}\,}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84zAe2yjG0nqnFyqnCztw3zDnCo2nFntGOatzBzDrFnDG3otK3aDFC)
![{\displaystyle \cosh ^{2}x={\frac {\cosh 2x+1}{2}}\,}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Bnqi0yjGOo2zCajG2zNmQata2z2iOnDm1aDi2zqrBoqw0otBBytC0)
![{\displaystyle \tanh ^{2}x={\frac {\cosh 2x-1}{\cosh 2x+1}}\,}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83ntK3aNs0oDoPzqo0oDm5nDe3yjaPoAe3zNhDaqrEoNzBztGPzNC5)
雙曲正切半形公式[编辑]
![{\displaystyle \sinh x={\frac {2\tanh {\frac {x}{2}}}{1-\tanh ^{2}{\frac {x}{2}}}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9EaDsQzgnFajdFzAo1zta2ytC4agi3zDK0nDaPnDC5zjrAnjrDnqrC)
![{\displaystyle \cosh x={\frac {1+\tanh ^{2}{\frac {x}{2}}}{1-\tanh ^{2}{\frac {x}{2}}}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83zNrFotdFoNK3yta5oDhEa2aPnjm5ngnDothEyqrDnjiOajC2ytFA)
![{\displaystyle \tanh x={\frac {2\tanh {\frac {x}{2}}}{1+\tanh ^{2}{\frac {x}{2}}}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80yjdDzAi4ztJCotGOa2sPytiOoArFati2ztw3ngoPzjm2zgrEztFB)
泰勒展開式[编辑]
![{\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)!}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81ata2nqrAyjK1ntG0zNw4a2i5yqzCaAnCnqeNnAs5yqrCaNFDntK2)
![{\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)!}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CngaPatKQnthDotJDz2iNyji1aAnDnAs4zjhEzDlCnqe0atFFnDvA)
![{\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|x\right|<{\frac {\pi }{2}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DytrDaqsQaNdDoNsOoDiQotrBajs4ajo2zteNoAzCzjs2zgoOatwN)
(罗朗级数)
![{\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|x\right|<{\frac {\pi }{2}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84oDa4agw1njJDz2aQnjw5a2eNzgi5njrCaNFCyqw2oDiOaNGPaAi2)
(罗朗级数)
其中
是第n項 伯努利數
是第n項 欧拉數
三角函數與雙曲函數的恆等式[编辑]
利用三角恒等式的指數定義和雙曲函數的指數定義即可求出下列恆等式:
所以
下表列出部分的三角函數與雙曲函數的恆等式:
三角函數
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雙曲函數
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![{\displaystyle \cosh ix={\tfrac {1}{2}}(e^{ix}+e^{-ix})=\cos x}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NaAeQnqhCngrBa2o2agsPa2a3njCQngnCnjo1oqo2otsNoAw3ngnD)
![{\displaystyle \sinh ix={\tfrac {1}{2}}(e^{ix}-e^{-ix})=i\sin x}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9FoDdEa2e4oDBBzDm0nqaPnje2zAhByjGPoNFFata0nAoPoDC0ati3)
![{\displaystyle \cosh(x+iy)=\cosh(x)\cos(y)+i\sinh(x)\sin(y)\,}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NaNJFnthCzDhDoNi3oDC0aNaOoNi2oAa5yjw1nDrCaDoOaDlDoqrF)
![{\displaystyle \sinh(x+iy)=\sinh(x)\cos(y)+i\cosh(x)\sin(y)\,}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80zNFAzqaOoqo5o2zCoto2nDzDntrAztnFaqzAzje2aNw4zDaPnjGO)
![{\displaystyle \tanh ix=i\tan x\,}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Dago1oqnFnAhFatvBotwPnDrBzNwNyqsNotdDytm1atwPzti2yjKO)
![{\displaystyle \cosh x=\cos ix\,}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81aNs2zAe5yjK3oDiPnDa0oDJEaDJEngoPz2w0oto4ngwNngi3oAeQ)
![{\displaystyle \sinh x=-i\sin ix\,}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DyqhDnAe4otnFoDiQnAzFoto0aqe0zDdAoDo5ntmOaAs5oqhEnAvD)
![{\displaystyle \tanh x=-i\tan ix\,}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84ztlDagnDztmOaDo4ntC3oqoQyjw1aAe3zDC4ntrAotwNoArDnjvC)
參考文獻[编辑]