雙曲函數恆等式

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。 雙曲函數基本恒等式如下： sinh ⁡ x = e x − e − x 2 {\displaystyle \sinh x={{e^{x}-e^{-x}} \over 2}} cosh ⁡ x = e x + e − x 2 {\displaystyle \cosh x={{e^{x}+e^{-x}} \over 2}} tanh ⁡ x = sinh ⁡ x cosh ⁡ x {\displaystyle \tanh x={{\sinh x} \over {\cosh x}}} coth ⁡ x = 1 tanh ⁡ x {\displaystyle \coth x={1 \over {\tanh x}}} s e c h x = 1 cosh ⁡ x {\displaystyle {\mathop {\rm {sech}} }x={1 \over {\cosh x}}} c s c h x = 1 sinh ⁡ 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） 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。 sinh ⁡ ( x + y )   = sinh ⁡ x cosh ⁡ y + cosh ⁡ x sinh ⁡ y {\displaystyle \sinh(x+y)\ =\sinh x\cosh y+\cosh x\sinh y\,} sinh ⁡ ( x − y )   = sinh ⁡ x cosh ⁡ y − cosh ⁡ x sinh ⁡ y {\displaystyle \sinh(x-y)\ =\sinh x\cosh y-\cosh x\sinh y\,} cosh ⁡ ( x + y )   = cosh ⁡ x cosh ⁡ y + sinh ⁡ x sinh ⁡ y {\displaystyle \cosh(x+y)\ =\cosh x\cosh y+\sinh x\sinh y\,} cosh ⁡ ( x − y )   = cosh ⁡ x cosh ⁡ y − sinh ⁡ x sinh ⁡ y {\displaystyle \cosh(x-y)\ =\cosh x\cosh y-\sinh x\sinh y\,} tanh ⁡ ( x + y )   = tanh ⁡ x + tanh ⁡ y 1 + tanh ⁡ x tanh ⁡ y {\displaystyle \tanh(x+y)\ ={\frac {\tanh x+\tanh y}{1+\tanh x\tanh y}}\,} tanh ⁡ ( x − y )   = tanh ⁡ x − tanh ⁡ y 1 − tanh ⁡ x tanh ⁡ y {\displaystyle \tanh(x-y)\ ={\frac {\tanh x-\tanh y}{1-\tanh x\tanh y}}\,} sinh ⁡ x + sinh ⁡ y   = 2 sinh ⁡ x + y 2 cosh ⁡ x − y 2 {\displaystyle \sinh x+\sinh y\ =2\sinh {\frac {x+y}{2}}\cosh {\frac {x-y}{2}}\,} sinh ⁡ x − sinh ⁡ y   = 2 cosh ⁡ x + y 2 sinh ⁡ x − y 2 {\displaystyle \sinh x-\sinh y\ =2\cosh {\frac {x+y}{2}}\sinh {\frac {x-y}{2}}\,} cosh ⁡ x + cosh ⁡ y   = 2 cosh ⁡ x + y 2 cosh ⁡ x − y 2 {\displaystyle \cosh x+\cosh y\ =2\cosh {\frac {x+y}{2}}\cosh {\frac {x-y}{2}}\,} cosh ⁡ x − cosh ⁡ y   = 2 sinh ⁡ x + y 2 sinh ⁡ x − y 2 {\displaystyle \cosh x-\cosh y\ =2\sinh {\frac {x+y}{2}}\sinh {\frac {x-y}{2}}\,} tanh ⁡ x + tanh ⁡ y   = sinh ⁡ ( x + y ) cosh ⁡ x cosh ⁡ y {\displaystyle \tanh x+\tanh y\ ={\frac {\sinh(x+y)}{\cosh x\cosh y}}\,} tanh ⁡ x − tanh ⁡ y   = sinh ⁡ ( x − y ) cosh ⁡ x cosh ⁡ y {\displaystyle \tanh x-\tanh y\ ={\frac {\sinh(x-y)}{\cosh x\cosh y}}\,} sinh ⁡ x sinh ⁡ y   = cosh ⁡ ( x + y ) − cosh ⁡ ( x − y ) 2 {\displaystyle \sinh x\sinh y\ ={\frac {\cosh(x+y)-\cosh(x-y)}{2}}\,} cosh ⁡ x cosh ⁡ y   = cosh ⁡ ( x + y ) + cosh ⁡ ( x − y ) 2 {\displaystyle \cosh x\cosh y\ ={\frac {\cosh(x+y)+\cosh(x-y)}{2}}\,} sinh ⁡ x cosh ⁡ y   = sinh ⁡ ( x + y ) + sinh ⁡ ( x − y ) 2 {\displaystyle \sinh x\cosh y\ ={\frac {\sinh(x+y)+\sinh(x-y)}{2}}\,} 二倍角公式： sinh ⁡ 2 x   = 2 sinh ⁡ x cosh ⁡ x {\displaystyle \sinh 2x\ =2\sinh x\cosh x\,} cosh ⁡ 2 x   = cosh 2 ⁡ x + sinh 2 ⁡ x = 2 cosh 2 ⁡ x − 1 = 2 sinh 2 ⁡ x + 1 {\displaystyle \cosh 2x\ =\cosh ^{2}x+\sinh ^{2}x=2\cosh ^{2}x-1=2\sinh ^{2}x+1\,} tanh ⁡ 2 x   = 2 tanh ⁡ x 1 + tanh 2 ⁡ x {\displaystyle \tanh 2x\ ={\frac {2\tanh x}{1+\tanh ^{2}x}}\,} 三倍角公式： sinh ⁡ 3 x   = 3 sinh ⁡ x + 4 sinh 3 ⁡ x {\displaystyle \sinh 3x\ =3\sinh x+4\sinh ^{3}x} cosh ⁡ 3 x   = 4 cosh 3 ⁡ x − 3 cosh ⁡ x {\displaystyle \cosh 3x\ =4\cosh ^{3}x-3\cosh x} sinh ⁡ x 2   = sgn ⁡ x cosh ⁡ x − 1 2 {\displaystyle \sinh {\frac {x}{2}}\ =\operatorname {sgn} x{\sqrt {\frac {\cosh x-1}{2}}}} cosh ⁡ x 2   = cosh ⁡ x + 1 2 {\displaystyle \cosh {\frac {x}{2}}\ ={\sqrt {\frac {\cosh x+1}{2}}}} tanh ⁡ x 2   = cosh ⁡ x − 1 sinh ⁡ x   = sinh ⁡ x 1 + cosh ⁡ x {\displaystyle \tanh {\frac {x}{2}}\ ={\frac {\cosh x-1}{\sinh x}}\ ={\frac {\sinh x}{1+\cosh x}}\,} sinh 2 ⁡ x = cosh ⁡ 2 x − 1 2 {\displaystyle \sinh ^{2}x={\frac {\cosh 2x-1}{2}}\,} cosh 2 ⁡ x = cosh ⁡ 2 x + 1 2 {\displaystyle \cosh ^{2}x={\frac {\cosh 2x+1}{2}}\,} tanh 2 ⁡ x = cosh ⁡ 2 x − 1 cosh ⁡ 2 x + 1 {\displaystyle \tanh ^{2}x={\frac {\cosh 2x-1}{\cosh 2x+1}}\,} sinh ⁡ x = 2 tanh ⁡ x 2 1 − tanh 2 ⁡ x 2 {\displaystyle \sinh x={\frac {2\tanh {\frac {x}{2}}}{1-\tanh ^{2}{\frac {x}{2}}}}} cosh ⁡ x = 1 + tanh 2 ⁡ x 2 1 − tanh 2 ⁡ x 2 {\displaystyle \cosh x={\frac {1+\tanh ^{2}{\frac {x}{2}}}{1-\tanh ^{2}{\frac {x}{2}}}}} tanh ⁡ x = 2 tanh ⁡ x 2 1 + tanh 2 ⁡ x 2 {\displaystyle \tanh x={\frac {2\tanh {\frac {x}{2}}}{1+\tanh ^{2}{\frac {x}{2}}}}} sinh ⁡ x = x + x 3 3 ! + x 5 5 ! + x 7 7 ! + ⋯ = ∑ n = 0 ∞ x 2 n + 1 ( 2 n + 1 ) ! {\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)!}}} cosh ⁡ x = 1 + x 2 2 ! + x 4 4 ! + x 6 6 ! + ⋯ = ∑ n = 0 ∞ x 2 n ( 2 n ) ! {\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)!}}} tanh ⁡ x = x − x 3 3 + 2 x 5 15 − 17 x 7 315 + ⋯ = ∑ n = 1 ∞ 2 2 n ( 2 2 n − 1 ) B 2 n x 2 n − 1 ( 2 n ) ! , | x | < π 2 {\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}}} coth ⁡ x = 1 x + x 3 − x 3 45 + 2 x 5 945 + ⋯ = 1 x + ∑ n = 1 ∞ 2 2 n B 2 n x 2 n − 1 ( 2 n ) ! , 0 < | x | < π {\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) sech x = 1 − x 2 2 + 5 x 4 24 − 61 x 6 720 + ⋯ = ∑ n = 0 ∞ E 2 n x 2 n ( 2 n ) ! , | x | < π 2 {\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}}} csch x = 1 x − x 6 + 7 x 3 360 − 31 x 5 15120 + ⋯ = 1 x + ∑ n = 1 ∞ 2 ( 1 − 2 2 n − 1 ) B 2 n x 2 n − 1 ( 2 n ) ! , 0 < | x | < π {\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 B n {\displaystyle B_{n}\,} 是第n項 伯努利數 E n {\displaystyle E_{n}\,} 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（英语：Hyperbolic_function#Hyperbolic_functions_for_complex_numbers）即可求出下列恆等式: e i x = cos ⁡ x + i sin ⁡ x , e − i x = cos ⁡ x − i sin ⁡ x {\displaystyle e^{ix}=\cos x+i\;\sin x\qquad ,\;e^{-ix}=\cos x-i\;\sin x} e x = cosh ⁡ x + sinh ⁡ x , e − x = cosh ⁡ x − sinh ⁡ x {\displaystyle e^{x}=\cosh x+\sinh x\!\qquad ,\;e^{-x}=\cosh x-\sinh x\!} 所以 cosh ⁡ i x = 1 2 ( e i x + e − i x ) = cos ⁡ x {\displaystyle \cosh ix={\tfrac {1}{2}}(e^{ix}+e^{-ix})=\cos x} sinh ⁡ i x = 1 2 ( e i x − e − i x ) = i sin ⁡ 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: 其他恆等式: cosh ⁡ i x = 1 2 ( e i x + e − i x ) = cos ⁡ x {\displaystyle \cosh ix={\tfrac {1}{2}}(e^{ix}+e^{-ix})=\cos x} sinh ⁡ i x = 1 2 ( e i x − e − i x ) = i sin ⁡ x {\displaystyle \sinh ix={\tfrac {1}{2}}(e^{ix}-e^{-ix})=i\sin x} cosh ⁡ ( x + i y ) = cosh ⁡ ( x ) cos ⁡ ( y ) + i sinh ⁡ ( x ) sin ⁡ ( y ) {\displaystyle \cosh(x+iy)=\cosh(x)\cos(y)+i\sinh(x)\sin(y)\,} sinh ⁡ ( x + i y ) = sinh ⁡ ( x ) cos ⁡ ( y ) + i cosh ⁡ ( x ) sin ⁡ ( y ) {\displaystyle \sinh(x+iy)=\sinh(x)\cos(y)+i\cosh(x)\sin(y)\,} tanh ⁡ i x = i tan ⁡ x {\displaystyle \tanh ix=i\tan x\,} cosh ⁡ x = cos ⁡ i x {\displaystyle \cosh x=\cos ix\,} sinh ⁡ x = − i sin ⁡ i x {\displaystyle \sinh x=-i\sin ix\,} tanh ⁡ x = − i tan ⁡ i x {\displaystyle \tanh x=-i\tan ix\,} 三角函數恆等式 雙曲函數 雙曲線 三角函數 三角形 數學基本公式手冊 九章出版社 ISBN 957-603-010-2