雙曲函數恆等式

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。

符号

	函数		倒數函数
	全寫	簡寫	全寫	簡寫
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

基本關係

sinh, cosh 和 tanh

csch, sech 和 coth

雙曲函數基本恒等式如下：

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

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

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

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

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

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

其他函數的基本關係

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	函數	值
雙曲正矢, hyperbolic versine	${\displaystyle \operatorname {versinh} (x)}$ ${\displaystyle \operatorname {vsnh} (x)}$	${\displaystyle \cosh x-1}$
雙曲餘矢, hyperbolic coversine	${\displaystyle \operatorname {coversinh} (x)}$ ${\displaystyle \operatorname {cvsh} (x)}$	${\displaystyle \sinh x-1}$
雙曲半正矢 , hyperbolic haversine	${\displaystyle \operatorname {haversinh} (x)}$	${\displaystyle {\frac {\operatorname {versinh} (x)}{2}}}$
雙曲半餘矢 , hyperbolic hacoversine	${\displaystyle \operatorname {hacoversinh} (x)}$	${\displaystyle {\frac {\operatorname {cvsh} (x)}{2}}}$
雙曲外正割 , hyperbolic exsecant	${\displaystyle \operatorname {exsech} (x)}$	${\displaystyle 1-\operatorname {sech} (x)}$
雙曲外餘割 , hyperbolic excosecant	${\displaystyle \operatorname {excsch} (x)}$	${\displaystyle 1-\operatorname {csch} (x)}$

和角公式

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

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

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

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

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

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

和差化積公式

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

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

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

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

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

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

積化和差公式

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

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

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

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

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

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

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

雙曲正切半形公式

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

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

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

泰勒展開式

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

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

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

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

${\displaystyle E_{n}\,}$ 是第n項 欧拉數

三角函數與雙曲函數的恆等式

利用三角恒等式的指數定義和雙曲函數的指數定義（英语：Hyperbolic_function#Hyperbolic_functions_for_complex_numbers）即可求出下列恆等式:

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

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

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數與雙曲函數的恆等式:

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

• 其他恆等式:

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

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

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

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

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

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

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

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

參見

• 三角函數恆等式
• 雙曲函數
• 雙曲線
• 三角函數
• 三角形

參考文獻

• 數學基本公式手冊 九章出版社 ISBN 957-603-010-2