基本定义 编辑 歷史 编辑 虛數圓角定義 编辑 與三角函數的類比 编辑 恆等式 编辑 双曲函数的導數 编辑
d d x sinh x = cosh x d d x cosh x = sinh x d d x tanh x = 1 − tanh 2 x = sech 2 x = 1 cosh 2 x d d x coth x = 1 − coth 2 x = − csch 2 x = − 1 sinh 2 x x ≠ 0 d d x sech x = − tanh x sech x d d x csch x = − coth x csch x x ≠ 0 {\displaystyle {\begin{aligned}{\frac {d}{dx}}\sinh x&=\cosh x\\{\frac {d}{dx}}\cosh x&=\sinh x\\{\frac {d}{dx}}\tanh x&=1-\tanh ^{2}x=\operatorname {sech} ^{2}x={\frac {1}{\cosh ^{2}x}}\\{\frac {d}{dx}}\coth x&=1-\coth ^{2}x=-\operatorname {csch} ^{2}x=-{\frac {1}{\sinh ^{2}x}}&&x\neq 0\\{\frac {d}{dx}}\operatorname {sech} x&=-\tanh x\operatorname {sech} x\\{\frac {d}{dx}}\operatorname {csch} x&=-\coth x\operatorname {csch} x&&x\neq 0\end{aligned}}}
双曲函数的泰勒展開式 编辑 雙曲函數也可以以泰勒級數 展開:
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 {\displaystyle n} 項伯努利數 E n {\displaystyle E_{n}} 是第 n {\displaystyle n} 項欧拉數 無限積與連續分數形式 编辑 双曲函数的积分 编辑 ∫ sinh c x d x = 1 c cosh c x + C {\displaystyle \int \sinh cx\,\mathrm {d} x={\frac {1}{c}}\cosh cx+C} ∫ cosh c x d x = 1 c sinh c x + C {\displaystyle \int \cosh cx\,\mathrm {d} x={\frac {1}{c}}\sinh cx+C} ∫ tanh c x d x = 1 c ln ( cosh c x ) + C {\displaystyle \int \tanh cx\,\mathrm {d} x={\frac {1}{c}}\ln(\cosh cx)+C} ∫ coth c x d x = 1 c ln | sinh c x | + C {\displaystyle \int \coth cx\,\mathrm {d} x={\frac {1}{c}}\ln \left|\sinh cx\right|+C} ∫ sech c x d x = 1 c arctan ( sinh c x ) + C {\displaystyle \int \operatorname {sech} cx\,\mathrm {d} x={\frac {1}{c}}\arctan(\sinh cx)+C} ∫ csch c x d x = 1 c ln | tanh c x 2 | + C {\displaystyle \int \operatorname {csch} cx\,\mathrm {d} x={\frac {1}{c}}\ln \left|\tanh {\frac {cx}{2}}\right|+C} 與指數函數的關係 编辑 複數的雙曲函數 编辑 反双曲函数 编辑 反双曲函数 是双曲函数的反函数 。它们的定义为:
arsinh ( x ) = ln ( x + x 2 + 1 ) arcosh ( x ) = ln ( x + x 2 − 1 ) ; x ≥ 1 artanh ( x ) = 1 2 ln ( 1 + x 1 − x ) ; | x | < 1 arcoth ( x ) = 1 2 ln ( x + 1 x − 1 ) ; | x | > 1 arsech ( x ) = ln ( 1 x + 1 − x 2 x ) ; 0 < x ≤ 1 arcsch ( x ) = ln ( 1 x + 1 + x 2 | x | ) ; x ≠ 0 {\displaystyle {\begin{aligned}\operatorname {arsinh} (x)&=\ln \left(x+{\sqrt {x^{2}+1}}\right)\\\operatorname {arcosh} (x)&=\ln \left(x+{\sqrt {x^{2}-1}}\right);x\geq 1\\\operatorname {artanh} (x)&={\frac {1}{2}}\ln \left({\frac {1+x}{1-x}}\right);\left|x\right|<1\\\operatorname {arcoth} (x)&={\frac {1}{2}}\ln \left({\frac {x+1}{x-1}}\right);\left|x\right|>1\\\operatorname {arsech} (x)&=\ln \left({\frac {1}{x}}+{\frac {\sqrt {1-x^{2}}}{x}}\right);0<x\leq 1\\\operatorname {arcsch} (x)&=\ln \left({\frac {1}{x}}+{\frac {\sqrt {1+x^{2}}}{\left|x\right|}}\right);x\neq 0\end{aligned}}} 参考文献 编辑 ^ 1.0 1.1 1.2 Weisstein, Eric W. (编). Hyperbolic Functions . at MathWorld --A Wolfram Web Resource. Wolfram Research, Inc. [2020-08-29 ] . (原始内容 存档于2022-05-21) (英语) . ^ Eves, Howard, Foundations and Fundamental Concepts of Mathematics , Courier Dover Publications: 59, 2012, ISBN 9780486132204 , We also owe to Lambert the first systematic development of the theory of hyperbolic functions and, indeed, our present notation for these functions. ^ Ratcliffe, John, Foundations of Hyperbolic Manifolds , Graduate Texts in Mathematics 149 , Springer: 99, 2006 [2014-03-27 ] , ISBN 9780387331973 , (原始内容存档 于2014-01-12), That the area of a hyperbolic triangle is proportional to its angle defect first appeared in Lambert's monograph Theorie der Parallellinien , which was published posthumously in 1786. ^ Augustus De Morgan (1849) Trigonometry and Double Algebra (页面存档备份 ,存于互联网档案馆 ), Chapter VI: "On the connection of common and hyperbolic trigonometry"^ G. Osborn, Mnemonic for hyperbolic formulae [失效連結 ] , The Mathematical Gazette, p. 189, volume 2, issue 34, July 1902 参见 编辑