双曲函数恒等式

在数学中，双曲函数恒等式是对出现的变量的所有值都为实的涉及到双曲函数的等式。这些恒等式在表达式中有些双曲函数需要简化的时候是很有用的。双曲函数的恒等式有的与三角恒等式类似。就如同三角函数，他有一个重要应用是非双曲函数的积分：一个常用技巧是首先使用换元积分法，规则与使用三角函数的代换规则类似，则通过双曲函数恒等式可简化结果的积分。

双曲扇形a的很多双曲函数可以在几何上依据以O为中心的双曲线来构造。

符号

	函数		倒数函数
	全写	简写	全写	简写
函数	hyperbolic sine	sinh	hyperbolic cosecant	csch
反函数	inverse hyperbolic sine	arcsinh	inverse hyperbolic cosecant	arccsch
函数	hyperbolic cosine	cosh	hyperbolic secant	sech
反函数	inverse hyperbolic cosine	arccosh	inverse hyperbolic secant	arcsech
函数	hyperbolic tangent	tanh	hyperbolic cotangent	coth
反函数	inverse hyperbolic tangent	arctanh	inverse hyperbolic cotangent	arccoth

基本关系

sinh, cosh 和 tanh

csch, sech 和 coth

双曲函数基本恒等式如下：

${\displaystyle \cosh ^{2}x-\sinh ^{2}x=1\,}$

${\displaystyle \tanh x\cdot \coth x\,=1}$

${\displaystyle 1\,-\tanh ^{2}x=\operatorname {sech} ^{2}x}$

${\displaystyle \coth ^{2}x-1\,=\operatorname {csch} ^{2}x}$

• ${\displaystyle \sinh x={{e^{x}-e^{-x}} \over 2}}$
• ${\displaystyle \cosh x={{e^{x}+e^{-x}} \over 2}}$
• ${\displaystyle \tanh x={{\sinh x} \over {\cosh x}}}$
• ${\displaystyle \coth x={1 \over {\tanh x}}}$
• ${\displaystyle {\mathop {\rm {sech}} }x={1 \over {\cosh x}}}$
• ${\displaystyle {\mathop {\rm {csch}} }x={1 \over {\sinh x}}}$

就如同三角函数，由上面的平方关系加上双曲函数的基本定义，可以导出下面的表格，即每个双曲函数都可以用其他五个表达。（严谨地说，所有根号前都应根据实际情况添加正负号）

函数	sinh	cosh	tanh	coth	sech	csch
${\displaystyle \sinh x}$	${\displaystyle \sinh x\ }$	${\displaystyle \operatorname {sgn} x{\sqrt {\cosh ^{2}x-1}}}$	${\displaystyle {\frac {\tanh x}{\sqrt {1-\tanh ^{2}x}}}}$	${\displaystyle {\frac {\operatorname {sgn} x}{\sqrt {\coth ^{2}x-1}}}}$	${\displaystyle \operatorname {sgn}(x){\frac {\sqrt {1-\operatorname {sech} ^{2}(x)}}{\operatorname {sech} (x)}}}$	${\displaystyle {\frac {1}{\operatorname {csch} (x)}}}$
${\displaystyle \cosh x}$	${\displaystyle {\sqrt {1+\sinh ^{2}x}}}$	${\displaystyle \cosh x\ }$	${\displaystyle {\frac {1}{\sqrt {1-\tanh ^{2}x}}}}$	${\displaystyle \,{\frac {\left|\coth(x)\right|}{\sqrt {\coth ^{2}(x)-1}}}}$	${\displaystyle \,{\frac {1}{\operatorname {sech} (x)}}}$	${\displaystyle \,{\frac {\sqrt {1+\operatorname {csch} ^{2}(x)}}{\left|\operatorname {csch} (x)\right|}}}$
${\displaystyle \tanh x}$	${\displaystyle {\frac {\sinh x}{\sqrt {1+\sinh ^{2}x}}}}$	${\displaystyle {\frac {\operatorname {sgn} x{\sqrt {\cosh ^{2}x-1}}}{\cosh x}}}$	${\displaystyle \tanh x\ }$	${\displaystyle {\frac {1}{\coth x}}}$	${\displaystyle \,\operatorname {sgn}(x){\sqrt {1-\operatorname {sech} ^{2}(x)}}}$	${\displaystyle \,{\frac {\operatorname {sgn}(x)}{\sqrt {1+\operatorname {csch} ^{2}(x)}}}}$
${\displaystyle \coth x}$	${\displaystyle {{\sqrt {1+\sinh ^{2}x}} \over \sinh x}}$	${\displaystyle {\cosh x \over \operatorname {sgn} x{\sqrt {\cosh ^{2}x-1}}}}$	${\displaystyle {1 \over \tanh x}}$	${\displaystyle \coth x\ }$	${\displaystyle \,{\frac {\operatorname {sgn}(x)}{\sqrt {1-\operatorname {sech} ^{2}(x)}}}}$	${\displaystyle \,\operatorname {sgn}(x){\sqrt {1+\operatorname {csch} ^{2}(x)}}}$
${\displaystyle \operatorname {sech} x}$	${\displaystyle {1 \over {\sqrt {1+\sinh ^{2}x}}}}$	${\displaystyle {1 \over \cosh \theta }}$	${\displaystyle {\sqrt {1-\tanh ^{2}x}}}$	${\displaystyle \,{\frac {\sqrt {\coth ^{2}(x)-1}}{\left|\coth(x)\right|}}}$	${\displaystyle \operatorname {sech} x\ }$	${\displaystyle \,{\frac {\left|\operatorname {csch} (x)\right|}{\sqrt {1+\operatorname {csch} ^{2}(x)}}}}$
${\displaystyle \operatorname {csch} x}$	${\displaystyle {1 \over \sinh x}}$	${\displaystyle {\frac {\operatorname {sgn} x}{\sqrt {\cosh ^{2}x-1}}}}$	${\displaystyle {\frac {\sqrt {1-\tanh ^{2}x}}{\tanh x}}}$	${\displaystyle \,\operatorname {sgn}(x){\sqrt {\coth ^{2}(x)-1}}}$	${\displaystyle \,\operatorname {sgn}(x){\frac {\operatorname {sech} (x)}{\sqrt {1-\operatorname {sech} ^{2}(x)}}}}$	${\displaystyle \operatorname {csch} x\ }$

其他函数的基本关系

三角函数还有正矢、余矢、半正矢、半余矢、外正割、外余割等函数，利用他们的定义也可以导出双曲函数。

名称	函数	值
双曲正矢, hyperbolic versine	${\displaystyle \operatorname {versinh} (x)}$ ${\displaystyle \operatorname {vsnh} (x)}$	${\displaystyle \cosh x-1}$
双曲余矢, hyperbolic coversine	${\displaystyle \operatorname {coversinh} (x)}$ ${\displaystyle \operatorname {cvsh} (x)}$	${\displaystyle \sinh x-1}$
双曲半正矢 , hyperbolic haversine	${\displaystyle \operatorname {haversinh} (x)}$	${\displaystyle {\frac {\operatorname {versinh} (x)}{2}}}$
双曲半余矢 , hyperbolic hacoversine	${\displaystyle \operatorname {hacoversinh} (x)}$	${\displaystyle {\frac {\operatorname {cvsh} (x)}{2}}}$
双曲外正割 , hyperbolic exsecant	${\displaystyle \operatorname {exsech} (x)}$	${\displaystyle 1-\operatorname {sech} (x)}$
双曲外余割 , hyperbolic excosecant	${\displaystyle \operatorname {excsch} (x)}$	${\displaystyle 1-\operatorname {csch} (x)}$

和角公式

${\displaystyle \sinh(x+y)\ =\sinh x\cosh y+\cosh x\sinh y\,}$

${\displaystyle \sinh(x-y)\ =\sinh x\cosh y-\cosh x\sinh y\,}$

${\displaystyle \cosh(x+y)\ =\cosh x\cosh y+\sinh x\sinh y\,}$

${\displaystyle \cosh(x-y)\ =\cosh x\cosh y-\sinh x\sinh y\,}$

${\displaystyle \tanh(x+y)\ ={\frac {\tanh x+\tanh y}{1+\tanh x\tanh y}}\,}$

${\displaystyle \tanh(x-y)\ ={\frac {\tanh x-\tanh y}{1-\tanh x\tanh y}}\,}$

和差化积公式

${\displaystyle \sinh x+\sinh y\ =2\sinh {\frac {x+y}{2}}\cosh {\frac {x-y}{2}}\,}$

${\displaystyle \sinh x-\sinh y\ =2\cosh {\frac {x+y}{2}}\sinh {\frac {x-y}{2}}\,}$

${\displaystyle \cosh x+\cosh y\ =2\cosh {\frac {x+y}{2}}\cosh {\frac {x-y}{2}}\,}$

${\displaystyle \cosh x-\cosh y\ =2\sinh {\frac {x+y}{2}}\sinh {\frac {x-y}{2}}\,}$

${\displaystyle \tanh x+\tanh y\ ={\frac {\sinh(x+y)}{\cosh x\cosh y}}\,}$

${\displaystyle \tanh x-\tanh y\ ={\frac {\sinh(x-y)}{\cosh x\cosh y}}\,}$

积化和差公式

${\displaystyle \sinh x\sinh y\ ={\frac {\cosh(x+y)-\cosh(x-y)}{2}}\,}$

${\displaystyle \cosh x\cosh y\ ={\frac {\cosh(x+y)+\cosh(x-y)}{2}}\,}$

${\displaystyle \sinh x\cosh y\ ={\frac {\sinh(x+y)+\sinh(x-y)}{2}}\,}$

倍角公式

• 二倍角公式：

${\displaystyle \sinh 2x\ =2\sinh x\cosh x\,}$

${\displaystyle \cosh 2x\ =\cosh ^{2}x+\sinh ^{2}x=2\cosh ^{2}x-1=2\sinh ^{2}x+1\,}$

${\displaystyle \tanh 2x\ ={\frac {2\tanh x}{1+\tanh ^{2}x}}\,}$

• 三倍角公式：

${\displaystyle \sinh 3x\ =3\sinh x+4\sinh ^{3}x}$

${\displaystyle \cosh 3x\ =4\cosh ^{3}x-3\cosh x}$

半形公式

${\displaystyle \sinh {\frac {x}{2}}\ =\operatorname {sgn} x{\sqrt {\frac {\cosh x-1}{2}}}}$

${\displaystyle \cosh {\frac {x}{2}}\ ={\sqrt {\frac {\cosh x+1}{2}}}}$

${\displaystyle \tanh {\frac {x}{2}}\ ={\frac {\cosh x-1}{\sinh x}}\ ={\frac {\sinh x}{1+\cosh x}}\,}$

幂简约公式

${\displaystyle \sinh ^{2}x={\frac {\cosh 2x-1}{2}}\,}$

${\displaystyle \cosh ^{2}x={\frac {\cosh 2x+1}{2}}\,}$

${\displaystyle \tanh ^{2}x={\frac {\cosh 2x-1}{\cosh 2x+1}}\,}$

双曲正切半形公式

${\displaystyle \sinh x={\frac {2\tanh {\frac {x}{2}}}{1-\tanh ^{2}{\frac {x}{2}}}}}$

${\displaystyle \cosh x={\frac {1+\tanh ^{2}{\frac {x}{2}}}{1-\tanh ^{2}{\frac {x}{2}}}}}$

${\displaystyle \tanh x={\frac {2\tanh {\frac {x}{2}}}{1+\tanh ^{2}{\frac {x}{2}}}}}$

泰勒展开式

${\displaystyle \sinh x=x+{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}+{\frac {x^{7}}{7!}}+\cdots =\sum _{n=0}^{\infty }{\frac {x^{2n+1}}{(2n+1)!}}}$

${\displaystyle \cosh x=1+{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}+{\frac {x^{6}}{6!}}+\cdots =\sum _{n=0}^{\infty }{\frac {x^{2n}}{(2n)!}}}$

${\displaystyle \tanh x=x-{\frac {x^{3}}{3}}+{\frac {2x^{5}}{15}}-{\frac {17x^{7}}{315}}+\cdots =\sum _{n=1}^{\infty }{\frac {2^{2n}(2^{2n}-1)B_{2n}x^{2n-1}}{(2n)!}},\left|x\right|<{\frac {\pi }{2}}}$

${\displaystyle \coth x={\frac {1}{x}}+{\frac {x}{3}}-{\frac {x^{3}}{45}}+{\frac {2x^{5}}{945}}+\cdots ={\frac {1}{x}}+\sum _{n=1}^{\infty }{\frac {2^{2n}B_{2n}x^{2n-1}}{(2n)!}},0<\left|x\right|<\pi }$ (罗朗级数)

${\displaystyle \operatorname {sech} \,x=1-{\frac {x^{2}}{2}}+{\frac {5x^{4}}{24}}-{\frac {61x^{6}}{720}}+\cdots =\sum _{n=0}^{\infty }{\frac {E_{2n}x^{2n}}{(2n)!}},\left|x\right|<{\frac {\pi }{2}}}$

${\displaystyle \operatorname {csch} \,x={\frac {1}{x}}-{\frac {x}{6}}+{\frac {7x^{3}}{360}}-{\frac {31x^{5}}{15120}}+\cdots ={\frac {1}{x}}+\sum _{n=1}^{\infty }{\frac {2(1-2^{2n-1})B_{2n}x^{2n-1}}{(2n)!}},0<\left|x\right|<\pi }$ (罗朗级数)

其中

${\displaystyle B_{n}\,}$ 是第n项 伯努利数

${\displaystyle E_{n}\,}$ 是第n项 欧拉数

三角函数与双曲函数的恒等式

利用三角恒等式的指数定义和双曲函数的指数定义（英语：Hyperbolic_function#Hyperbolic_functions_for_complex_numbers）即可求出下列恒等式:

${\displaystyle e^{ix}=\cos x+i\;\sin x\qquad ,\;e^{-ix}=\cos x-i\;\sin x}$

${\displaystyle e^{x}=\cosh x+\sinh x\!\qquad ,\;e^{-x}=\cosh x-\sinh x\!}$

所以

${\displaystyle \cosh ix={\tfrac {1}{2}}(e^{ix}+e^{-ix})=\cos x}$

${\displaystyle \sinh ix={\tfrac {1}{2}}(e^{ix}-e^{-ix})=i\sin x}$

下表列出部分的三角函数与双曲函数的恒等式:

三角函数	双曲函数
${\displaystyle \sin \theta =-i\sinh {i\theta }\,}$	${\displaystyle \sinh {\theta }=i\sin {(-i\theta )}\,}$
${\displaystyle \cos {\theta }=\cosh {i\theta }\,}$	${\displaystyle \cosh {\theta }=\cos {(-i\theta )}\,}$
${\displaystyle \tan \theta ={\frac {\tanh {i\theta }}{i}}\,}$	${\displaystyle \tanh {\theta }=i\tan {(-i\theta )}\,}$
${\displaystyle \cot {\theta }=i\coth {i\theta }\,}$	${\displaystyle \coth \theta ={\frac {\cot {(-i\theta )}}{i}}\,}$
${\displaystyle \sec {\theta }=\operatorname {sech} {\,i\theta }\,}$	${\displaystyle \operatorname {sech} {\theta }=\sec {(-i\theta )}\,}$
${\displaystyle \csc {\theta }=i\;\operatorname {csch} {\,i\theta }\,}$	${\displaystyle \operatorname {csch} \theta ={\frac {\csc {(-i\theta )}}{i}}\,}$

• 其他恒等式:

${\displaystyle \cosh ix={\tfrac {1}{2}}(e^{ix}+e^{-ix})=\cos x}$

${\displaystyle \sinh ix={\tfrac {1}{2}}(e^{ix}-e^{-ix})=i\sin x}$

${\displaystyle \cosh(x+iy)=\cosh(x)\cos(y)+i\sinh(x)\sin(y)\,}$

${\displaystyle \sinh(x+iy)=\sinh(x)\cos(y)+i\cosh(x)\sin(y)\,}$

${\displaystyle \tanh ix=i\tan x\,}$

${\displaystyle \cosh x=\cos ix\,}$

${\displaystyle \sinh x=-i\sin ix\,}$

${\displaystyle \tanh x=-i\tan ix\,}$

参见

• 三角函数恒等式
• 双曲函数
• 双曲线
• 三角函数
• 三角形

参考文献

• 数学基本公式手册 九章出版社 ISBN 957-603-010-2