# 雙曲函數

## 基本定義

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

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

## 歷史

• ${\displaystyle \cosh u={\frac {e^{u}+e^{-u}}{2}}}$
• ${\displaystyle \sinh u={\frac {e^{u}-e^{-u}}{2}}}$

## 虛數圓角定義

${\displaystyle \cos(ix)=\cosh(x),\quad }$    ${\displaystyle \quad \sin(ix)=i\sinh(x).}$

${\displaystyle e^{x}=\cosh x+\sinh x\!}$
${\displaystyle {\begin{array}{lcl}\cosh x=\sum _{n=0}^{\infty }{\frac {x^{2n}}{(2n)!}}&\sinh x=\sum _{n=0}^{\infty }{\frac {x^{2n+1}}{(2n+1)!}}\\\cos x=\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n}}{(2n)!}}&\sin x=\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{(2n+1)!}}\\\end{array}}}$

${\displaystyle \sinh x=-i\sin(ix)\!}$
${\displaystyle \cosh x=\cos(ix)\!}$
${\displaystyle \tanh x=-i\tan(ix)\!}$
${\displaystyle \coth x=i\cot(ix)\!}$
${\displaystyle \operatorname {sech} x=\sec(ix)\!}$
${\displaystyle \operatorname {csch} x=i\csc(ix)\!}$

## 恆等式

${\displaystyle \cosh ^{2}x-\sinh ^{2}x=1\,}$
• 加法公式：
${\displaystyle \sinh(x+y)=\sinh x\cosh y+\cosh x\sinh y\,}$
${\displaystyle \cosh(x+y)=\cosh x\cosh y+\sinh x\sinh y\,}$
${\displaystyle \tanh(x+y)={\frac {\tanh x+\tanh y}{1+\tanh x\tanh y}}\,}$
• 二倍角公式：
${\displaystyle \sinh 2x\ =2\sinh x\cosh x\,}$
${\displaystyle \cosh 2x\ =\cosh ^{2}x+\sinh ^{2}x=2\cosh ^{2}x-1=2\sinh ^{2}x+1\,}$
• 半角公式：
${\displaystyle \cosh ^{2}{\frac {x}{2}}={\frac {\cosh x+1}{2}}}$
${\displaystyle \sinh ^{2}{\frac {x}{2}}={\frac {\cosh x-1}{2}}}$

• 三倍角公式：

${\displaystyle \sin 3x\ =3\sin x-4\sin ^{3}x}$
${\displaystyle \cos 3x\ =-3\cos x+4\cos ^{3}x}$

${\displaystyle \sinh 3x\ =3\sinh x+4\sinh ^{3}x}$
${\displaystyle \cosh 3x\ =-3\cosh x+4\cosh ^{3}x}$
• 差角公式：

## 雙曲函數的導數

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\sinh x=\cosh x\,}$
${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\cosh x=\sinh x\,}$
${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\tanh x=1-\tanh ^{2}x={\hbox{sech}}^{2}x={\frac {1}{\cosh ^{2}x}}\,}$

## 雙曲函數的泰勒展開式

${\displaystyle \sinh x=x+{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}+{\frac {x^{7}}{7!}}+\cdots =\sum _{n=0}^{\infty }{\frac {x^{2n+1}}{(2n+1)!}}}$
${\displaystyle \cosh x=1+{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}+{\frac {x^{6}}{6!}}+\cdots =\sum _{n=0}^{\infty }{\frac {x^{2n}}{(2n)!}}}$
${\displaystyle \tanh x=x-{\frac {x^{3}}{3}}+{\frac {2x^{5}}{15}}-{\frac {17x^{7}}{315}}+\cdots =\sum _{n=1}^{\infty }{\frac {2^{2n}(2^{2n}-1)B_{2n}x^{2n-1}}{(2n)!}},\left|x\right|<{\frac {\pi }{2}}}$
${\displaystyle \coth x={\frac {1}{x}}+{\frac {x}{3}}-{\frac {x^{3}}{45}}+{\frac {2x^{5}}{945}}+\cdots ={\frac {1}{x}}+\sum _{n=1}^{\infty }{\frac {2^{2n}B_{2n}x^{2n-1}}{(2n)!}},0<\left|x\right|<\pi }$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

${\displaystyle B_{n}\,}$是第n項伯努利數
${\displaystyle E_{n}\,}$是第n項歐拉數

## 雙曲函數的積分

${\displaystyle \int \sinh cx\,\mathrm {d} x={\frac {1}{c}}\cosh cx+C}$
${\displaystyle \int \cosh cx\,\mathrm {d} x={\frac {1}{c}}\sinh cx+C}$
${\displaystyle \int \tanh cx\,\mathrm {d} x={\frac {1}{c}}\ln(\cosh cx)+C}$
${\displaystyle \int \coth cx\,\mathrm {d} x={\frac {1}{c}}\ln(\sinh cx)+C}$
${\displaystyle \int \operatorname {sech} cx\,\mathrm {d} x={\frac {1}{c}}\arctan(\sinh cx)+C}$
${\displaystyle \int \operatorname {csch} cx\,\mathrm {d} x={\frac {1}{c}}\ln \left|\tanh {\frac {cx}{2}}\right|+C}$

## 與指數函數的關係

${\displaystyle e^{x}=\cosh x+\sinh x}$

${\displaystyle e^{-x}=\cosh x-\sinh x}$

## 複數的雙曲函數

{\displaystyle {\begin{aligned}e^{ix}&=\cos x+i\;\sin x\\e^{-ix}&=\cos x-i\;\sin x\end{aligned}}}

{\displaystyle {\begin{aligned}\cosh ix&={\frac {1}{2}}\left(e^{ix}+e^{-ix}\right)=\cos x\\\sinh ix&={\frac {1}{2}}\left(e^{ix}-e^{-ix}\right)=i\sin x\\\tanh ix&=i\tan x\\\end{aligned}}}
{\displaystyle {\begin{aligned}\cosh(x+iy)&=\cosh(x)\cos(y)+i\sinh(x)\sin(y)\\\sinh(x+iy)&=\sinh(x)\cos(y)+i\cosh(x)\sin(y)\\\end{aligned}}}
{\displaystyle {\begin{aligned}\cosh x&=\cos ix\\\sinh x&=-i\sin ix\\\tanh x&=-i\tan ix\end{aligned}}}

## 反雙曲函數

{\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

## 註釋與引用

1. ^ 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.
2. ^ Ratcliffe, John, Foundations of Hyperbolic Manifolds, Graduate Texts in Mathematics 149, Springer: 99, 2006, ISBN 9780387331973, 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.
3. ^ Augustus De Morgan (1849) Trigonometry and Double Algebra, Chapter VI: "On the connection of common and hyperbolic trigonometry"
4. ^ G. Osborn, Mnemonic for hyperbolic formulae, The Mathematical Gazette, p. 189, volume 2, issue 34, July 1902