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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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}}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/f1741241aebaa576869f5407f7f4b1bdf7ea5ffe)
![{\displaystyle \cosh x={{e^{x}+e^{-x}} \over 2}}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/8047c0dd01ecd050d17b883b137d91bc0b59f6f4)
![{\displaystyle \tanh x={{\sinh x} \over {\cosh x}}}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/66636332e791dc4d587e278e18572fcbeaa77f43)
![{\displaystyle \coth x={1 \over {\tanh x}}}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/d5c259abeb83e81f41b9d6b3767d0f981e830e8a)
![{\displaystyle {\mathop {\rm {sech}} }x={1 \over {\cosh x}}}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/dd8f8d203b1e386dadb0877874369d460fdfd7e8)
![{\displaystyle {\mathop {\rm {csch}} }x={1 \over {\sinh x}}}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/0d70e98f983a17b09209912626cc76f759624829)
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���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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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)}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/37925f7495e5d14b9bb2a88c3e9bc2a3e46d3d31)
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雙曲餘矢, hyperbolic coversine
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![{\displaystyle \operatorname {coversinh} (x)}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/04aa585e7a7c35641afbf4defddfe4f6c2880b85)
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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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![{\displaystyle \sinh(x+y)\ =\sinh x\cosh y+\cosh x\sinh y\,}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/acecdd9b304702ab60795ba7466eb74c22d95865)
![{\displaystyle \sinh(x-y)\ =\sinh x\cosh y-\cosh x\sinh y\,}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/40557c6ef3950d555d71e93e0a1875c52bfb99f9)
![{\displaystyle \cosh(x+y)\ =\cosh x\cosh y+\sinh x\sinh y\,}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/0fa3745d35e133ea8288cd3fde0227e202d3d20f)
![{\displaystyle \cosh(x-y)\ =\cosh x\cosh y-\sinh x\sinh y\,}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/0ebcbdec5483eeb0a0ed53a85063466313c9fffc)
![{\displaystyle \tanh(x+y)\ ={\frac {\tanh x+\tanh y}{1+\tanh x\tanh y}}\,}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/f6ffeb67c65fedb3bd13eb5944c9c177b70802de)
![{\displaystyle \tanh(x-y)\ ={\frac {\tanh x-\tanh y}{1-\tanh x\tanh y}}\,}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/42c3ccf7e7ab3d5386264af6573483da44dbdd3f)
![{\displaystyle \sinh x+\sinh y\ =2\sinh {\frac {x+y}{2}}\cosh {\frac {x-y}{2}}\,}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/547e80749b648e06552b02e8c15f8c9ab20458d0)
![{\displaystyle \sinh x-\sinh y\ =2\cosh {\frac {x+y}{2}}\sinh {\frac {x-y}{2}}\,}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/c5438b85441b61fa8afe979fb8c3537f51f941bf)
![{\displaystyle \cosh x+\cosh y\ =2\cosh {\frac {x+y}{2}}\cosh {\frac {x-y}{2}}\,}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/0edef61000fd53c97505ce07ff2dc40dd9335d23)
![{\displaystyle \cosh x-\cosh y\ =2\sinh {\frac {x+y}{2}}\sinh {\frac {x-y}{2}}\,}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/c2418eb1bb2741d1cc52dc02f2625a2ba2cad874)
![{\displaystyle \tanh x+\tanh y\ ={\frac {\sinh(x+y)}{\cosh x\cosh y}}\,}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/b226bc5f622f2dddb85a1f00ef07f5bff28ac101)
![{\displaystyle \tanh x-\tanh y\ ={\frac {\sinh(x-y)}{\cosh x\cosh y}}\,}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/b9c8f1f13b5662375d6e895801ae9788a22899d4)
![{\displaystyle \sinh x\sinh y\ ={\frac {\cosh(x+y)-\cosh(x-y)}{2}}\,}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/3624af23f8cd39fed986a178f3cb2e11b6b1c8ec)
![{\displaystyle \cosh x\cosh y\ ={\frac {\cosh(x+y)+\cosh(x-y)}{2}}\,}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/e382f9a8ec535bf5dab7c43f2f0411292fc03dd7)
![{\displaystyle \sinh x\cosh y\ ={\frac {\sinh(x+y)+\sinh(x-y)}{2}}\,}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/7ada5f6be3129ae4d16f8425158e2a8e2a114beb)
![{\displaystyle \sinh 2x\ =2\sinh x\cosh x\,}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/e3cd93d9c40bb0e6e6cee1258fcc5465f63fd41f)
![{\displaystyle \cosh 2x\ =\cosh ^{2}x+\sinh ^{2}x=2\cosh ^{2}x-1=2\sinh ^{2}x+1\,}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/54eb58856a950bc60b6d5275dfc4e4af130fc76c)
![{\displaystyle \tanh 2x\ ={\frac {2\tanh x}{1+\tanh ^{2}x}}\,}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/cfefb15ea11fb38c89a41fd2fe4f70b77c9ac753)
![{\displaystyle \sinh 3x\ =3\sinh x+4\sinh ^{3}x}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/3939a39156480d2815dbca6c156f99ea7ef6fb3b)
![{\displaystyle \cosh 3x\ =4\cosh ^{3}x-3\cosh x}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/e09e238efb8e0a58b570d5e4b6b14de0ad49bcb6)
![{\displaystyle \sinh {\frac {x}{2}}\ =\operatorname {sgn} x{\sqrt {\frac {\cosh x-1}{2}}}}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/0f8ec72074616e52fdfc840459e77f370cd7f92d)
![{\displaystyle \cosh {\frac {x}{2}}\ ={\sqrt {\frac {\cosh x+1}{2}}}}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/2d1dc47f249d6d1410e6d8a4c3b45644e0f84ec9)
![{\displaystyle \tanh {\frac {x}{2}}\ ={\frac {\cosh x-1}{\sinh x}}\ ={\frac {\sinh x}{1+\cosh x}}\,}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/5c56de60b83cde2f2eb503e509b7fd73b5552e1c)
![{\displaystyle \sinh ^{2}x={\frac {\cosh 2x-1}{2}}\,}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/86b6884efa9fd54766dcfae8213e65af87a7728d)
![{\displaystyle \cosh ^{2}x={\frac {\cosh 2x+1}{2}}\,}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/eea4882ccd0867001367a2f052165eead4a9e994)
![{\displaystyle \tanh ^{2}x={\frac {\cosh 2x-1}{\cosh 2x+1}}\,}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/7e77354b615f4b09f27831bb771c1ebc3e581799)
![{\displaystyle \sinh x={\frac {2\tanh {\frac {x}{2}}}{1-\tanh ^{2}{\frac {x}{2}}}}}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/b2504fa02a6f55369980a7674f31f9945fd5ceed)
![{\displaystyle \cosh x={\frac {1+\tanh ^{2}{\frac {x}{2}}}{1-\tanh ^{2}{\frac {x}{2}}}}}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/775aa2ac77939b1b3c1d09dfca1b9ecd1209698f)
![{\displaystyle \tanh x={\frac {2\tanh {\frac {x}{2}}}{1+\tanh ^{2}{\frac {x}{2}}}}}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/482c6a857da823e1912bea116547df14064eb58e)
![{\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)!}}}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/5136eef875e847483a99cd2fdeb3fe99ed38ce76)
![{\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)!}}}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/ddc1170e1ca7c7a38152fcfe841b60deb418af4f)
![{\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}}}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/c95c1e032cc52b10a5e058066523bcd4564f2143)
(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}}}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/8b380d5d7c7c0d493b34a9d5d38d9d6b123812a6)
(罗朗级数)
其中
是第n項 伯努利數
是第n項 欧拉數
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}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/32b0eaddee3f60e13c7d90dfdd65af6a53bd7dfc)
![{\displaystyle \sinh ix={\tfrac {1}{2}}(e^{ix}-e^{-ix})=i\sin x}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/ab2b3b8b9e604ec1d266ae881c8a134ff1b94117)
![{\displaystyle \cosh(x+iy)=\cosh(x)\cos(y)+i\sinh(x)\sin(y)\,}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/337ae1d61cc17b94332c16bc9845f5d26220caea)
![{\displaystyle \sinh(x+iy)=\sinh(x)\cos(y)+i\cosh(x)\sin(y)\,}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/478f5c2af9ccda66f3ce5f56a1cf426348631d82)
![{\displaystyle \tanh ix=i\tan x\,}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/c0f5afafaa14ea41f5e7439e3a2c905141516872)
![{\displaystyle \cosh x=\cos ix\,}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/53566b9877b11f34b7b27bdf17d4a68dd3da7bb0)
![{\displaystyle \sinh x=-i\sin ix\,}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/c9acfb8a6ab10fcaa641b462fb69e022e9aabfdc)
![{\displaystyle \tanh x=-i\tan ix\,}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/850c0fc502268e97af08452b7698e5fa43becd4d)