# 古德曼函數

## 性質

{\displaystyle {\begin{aligned}{\rm {gd}}(x)&=\int _{0}^{x}{\frac {dt}{\cosh t}}\qquad -\infty

{\displaystyle {\begin{aligned}{\rm {gd}}(x)=\mathrm {arccot} \left(\mathrm {csch} \,x\right)\end{aligned}}\,\!}僅在arccot的值域設為${\displaystyle [-{\frac {\pi }{2}},{\frac {\pi }{2}}]}$時成立，參見反餘切。）

{\displaystyle {\begin{aligned}\sin \left({\mbox{gd}}x\right)&=\tanh x;&\quad \cos \left({\mbox{gd}}x\right)&={\mbox{sech}}x\\\tan \left({\mbox{gd}}x\right)&=\sinh x;&\quad \sec \left({\mbox{gd}}x\right)&=\cosh x\\\cot \left({\mbox{gd}}x\right)&={\mbox{csch}}x;&\quad \csc \left({\mbox{gd}}x\right)&=\coth x\\\tan \left({\frac {{\mbox{gd}}x}{2}}\right)&=\tanh {\frac {x}{2}};&\quad \cot \left({\frac {{\mbox{gd}}x}{2}}\right)&=\coth {\frac {x}{2}}\\\end{aligned}}\,\!}

## 反函數

{\displaystyle {\begin{aligned}{\mbox{arcgd}}x&={\rm {gd}}^{-1}x=\int _{0}^{x}{\frac {dt}{\cos t}}\qquad -\pi /2

{\displaystyle {\begin{aligned}\sinh \left({\mbox{gd}}^{-1}x\right)&=\tan x;&\quad \cosh \left({\mbox{gd}}^{-1}x\right)&=\sec x\\\tanh \left({\mbox{gd}}^{-1}x\right)&=\sin x;&\quad \;{\mbox{sech}}\left({\mbox{gd}}^{-1}x\right)&=\cos x\\\coth \left({\mbox{gd}}^{-1}x\right)&=\csc x;&\quad \,{\mbox{csch}}\left({\mbox{gd}}^{-1}x\right)&=\cot x\\\tanh \left({\frac {{\mbox{gd}}^{-1}x}{2}}\right)&=\tan {\frac {x}{2}};&\quad \,\coth \left({\frac {{\mbox{gd}}^{-1}x}{2}}\right)&=\cot {\frac {x}{2}}\\\end{aligned}}\,\!}

## 餘函數

{\displaystyle {\begin{aligned}{\mbox{cogd}}x&={\begin{cases}\int _{\infty }^{x}{\frac {dt}{\sinh t}}\qquad 0

{\displaystyle {\begin{aligned}\sinh \left({\mbox{cogd}}x\right)&={\mbox{csch}}x;&\quad \;\cosh \left({\mbox{cogd}}x\right)&=\coth \left|x\right|\\\tanh \left(\left|{\mbox{cogd}}x\right|\right)&={\mbox{sech}}x;&\quad \;{\mbox{sech}}\left({\mbox{cogd}}x\right)&=\tanh \left|x\right|\\\coth \left(\left|{\mbox{cogd}}x\right|\right)&=\cosh x;&\quad \,{\mbox{csch}}\left({\mbox{cogd}}x\right)&=\sinh x\\\end{aligned}}\,\!}

## 微分

{\displaystyle {\begin{aligned}{\frac {d}{dx}}{\mbox{gd}}x={\mbox{sech}}x;\quad {\frac {d}{dx}}{\mbox{arcgd}}x=\sec x;\quad {\frac {d}{dx}}{\mbox{cogd}}x=-{\mbox{csch}}\left|x\right|\\\end{aligned}}\,\!}

## 應用

${\displaystyle {\frac {\pi }{2}}-{\mbox{gd}}(x)}$

• 在使用麥卡托投影法的地圖，若以${\displaystyle y\,}$表示一個地點在地圖跟赤道的距離，則其緯度${\displaystyle \phi \,}$${\displaystyle y\,}$的關係為：
${\displaystyle \phi ={\mbox{gd}}(y)\,}$
• 古德曼函數在倒單擺的非週期解中出現。