雙曲函數恆等式

在數學中，雙曲函數恆等式是對出現的變量的所有值都為實的涉及到雙曲函數的等式。這些恆等式在表達式中有些雙曲函數需要簡化的時候是很有用的。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，他有一個重要應用是非雙曲函數的積分：一個常用技巧是首先使用換元積分法，規則與使用三角函數的代換規則類似，則通過雙曲函數恆等式可簡化結果的積分。

雙曲扇形a的很多雙曲函數可以在幾何上依據以O為中心的雙曲線來構造。

符號

	函數		倒數函數
	全寫	簡寫	全寫	簡寫
函數	hyperbolic sine	sinh	hyperbolic cosecant	csch
反函數	inverse hyperbolic sine	arcsinh	inverse hyperbolic cosecant	arccsch
函數	hyperbolic cosine	cosh	hyperbolic secant	sech
反函數	inverse hyperbolic cosine	arccosh	inverse hyperbolic secant	arcsech
函數	hyperbolic tangent	tanh	hyperbolic cotangent	coth
反函數	inverse hyperbolic tangent	arctanh	inverse hyperbolic cotangent	arccoth

基本關係

sinh, cosh 和 tanh

csch, sech 和 coth

雙曲函數基本恆等式如下：

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

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

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

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

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

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。（嚴謹地說，所有根號前都應根據實際情況添加正負號）

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

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

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

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

冪簡約公式

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

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

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

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

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

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

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

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

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

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

其中

${\displaystyle B_{n}\,}$ 是第n項 伯努利數

${\displaystyle E_{n}\,}$ 是第n項 歐拉數

三角函數與雙曲函數的恆等式

利用三角恆等式的指數定義和雙曲函數的指數定義（英語：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