古德曼函數

古德曼函數（Gudermannian function）是一個函數。它無須涉及複數便將三角函數和雙曲函數連繫起來。 古德曼函數的定義如下 g d ( x ) = ∫ 0 x d t cosh ⁡ t − ∞ < x < ∞ = arcsin ⁡ ( tanh ⁡ x ) = arctan ( sinh ⁡ x ) = a r c c s c ( coth ⁡ x ) = sgn ( x ) ⋅ a r c c o s ( s e c h x ) = sgn ( x ) ⋅ a r c s e c ( cosh ⁡ x ) = 2 arctan ⁡ ( e x ) − π 2 = π 2 − 2 arccot ⁡ ( e x ) = 2 arctan ⁡ ( tanh ⁡ x 2 ) = a r c c o t ( c s c h x ) {\displaystyle {\begin{aligned}{\rm {gd}}(x)&=\int _{0}^{x}{\frac {dt}{\cosh t}}\qquad -\infty <x<\infty \\&=\arcsin \left(\tanh x\right)={\mbox{arctan}}\left(\sinh x\right)=\mathrm {arccsc} \left(\coth x\right)\\&={\mbox{sgn}}(x)\cdot \mathrm {arccos} \left(\mathrm {sech} \,x\right)={\mbox{sgn}}(x)\cdot \mathrm {arcsec} \left(\cosh x\right)\\&=2\arctan(e^{x})-{\frac {\pi }{2}}={\frac {\pi }{2}}-2\operatorname {arccot}(e^{x})\\&=2\arctan \left(\tanh {\frac {x}{2}}\right)\\&=\mathrm {arccot} \left(\mathrm {csch} \,x\right)\\\end{aligned}}\,\!} （ g d ( x ) = a r c c o t ( c s c h x ) {\displaystyle {\begin{aligned}{\rm {gd}}(x)=\mathrm {arccot} \left(\mathrm {csch} \,x\right)\end{aligned}}\,\!} 僅在arccot的值域設為 [ − π 2 , π 2 ] {\displaystyle [-{\frac {\pi }{2}},{\frac {\pi }{2}}]} 時成立，參見反餘切。） 有以下恆等式： sin ⁡ ( gd x ) = tanh ⁡ x ; cos ⁡ ( gd x ) = sech x tan ⁡ ( gd x ) = sinh ⁡ x ; sec ⁡ ( gd x ) = cosh ⁡ x cot ⁡ ( gd x ) = csch x ; csc ⁡ ( gd x ) = coth ⁡ x tan ⁡ ( gd x 2 ) = tanh ⁡ x 2 ; cot ⁡ ( gd x 2 ) = coth ⁡ x 2 {\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}}\,\!} 古德曼函數之反函數的定義為： arcgd x = g d − 1 x = ∫ 0 x d t cos ⁡ t − π / 2 < x < π / 2 = a r c t a n h ( sin ⁡ x ) = a r c s i n h ( tan ⁡ x ) = a r c c o t h ( csc ⁡ x ) = a r c c s c h ( cot ⁡ x ) = sgn ( x ) ⋅ a r c c o s h ( sec ⁡ x ) = sgn ( x ) ⋅ a r c s e c h ( cos ⁡ x ) = 2 a r c t a n h ( tan ⁡ x 2 ) = ln ⁡ | sec ⁡ x ( 1 + sin ⁡ x ) | = ln ⁡ | tan ⁡ x + sec ⁡ x | = ln ⁡ | tan ⁡ ( π 4 + x 2 ) | = 1 2 ln ⁡ | 1 + sin ⁡ x 1 − sin ⁡ x | {\displaystyle {\begin{aligned}{\mbox{arcgd}}x&={\rm {gd}}^{-1}x=\int _{0}^{x}{\frac {dt}{\cos t}}\qquad -\pi /2<x<\pi /2\\&=\mathrm {arctanh} \,(\sin x)=\mathrm {arcsinh} \,(\tan x)\\&=\mathrm {arccoth} \,(\csc x)=\mathrm {arccsch} \,(\cot x)\\&={\mbox{sgn}}(x)\cdot \mathrm {arccosh} \,(\sec x)={\mbox{sgn}}(x)\cdot \mathrm {arcsech} \,(\cos x)\\&=2\mathrm {arctanh} \left(\tan {\frac {x}{2}}\right)\\&={}\ln \left|\sec x(1+\sin x)\right|\\&={}\ln \left|\tan x+\sec x\right|=\ln \left|\tan \left({\frac {\pi }{4}}+{\frac {x}{2}}\right)\right|\\&={}{\frac {1}{2}}\ln \left|{\frac {1+\sin x}{1-\sin x}}\right|\end{aligned}}\,\!} 有以下恆等式： sinh ⁡ ( gd − 1 x ) = tan ⁡ x ; cosh ⁡ ( gd − 1 x ) = sec ⁡ x tanh ⁡ ( gd − 1 x ) = sin ⁡ x ; sech ( gd − 1 x ) = cos ⁡ x coth ⁡ ( gd − 1 x ) = csc ⁡ x ; csch ( gd − 1 x ) = cot ⁡ x tanh ⁡ ( gd − 1 x 2 ) = tan ⁡ x 2 ; coth ⁡ ( gd − 1 x 2 ) = cot ⁡ x 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}}\,\!} 古德曼函數之餘函數的定義為： cogd x = { ∫ ∞ x d t sinh ⁡ t 0 < x < ∞ ∫ x − ∞ d t sinh ⁡ t − ∞ < x < 0 = − sgn ( x ) ⋅ ln ⁡ | tanh ⁡ x 2 | = sgn ( x ) ⋅ ln ⁡ | coth ⁡ x + csch x | = 2 artanh ( e − | x | ) ⋅ sgn ( x ) = 2 arcoth ( e | x | ) ⋅ sgn ( x ) = cogd − 1 x {\displaystyle {\begin{aligned}{\mbox{cogd}}x&={\begin{cases}\int _{\infty }^{x}{\frac {dt}{\sinh t}}\qquad 0<x<\infty \!\,\\\int _{x}^{-\infty }{\frac {dt}{\sinh t}}\qquad -\infty <x<0\!\,\end{cases}}\\&=-{\mbox{sgn}}(x)\cdot \ln \left|\tanh {x \over 2}\right|\\&={\mbox{sgn}}(x)\cdot \ln \left|\coth x+{\mbox{csch}}x\right|\\&=2{\mbox{artanh}}(e^{-\left|x\right|})\cdot {\mbox{sgn}}(x)=2{\mbox{arcoth}}(e^{\left|x\right|})\cdot {\mbox{sgn}}(x)\\&={\mbox{cogd}}^{-1}x\\\end{aligned}}\,\!} 有以下恆等式： sinh ⁡ ( cogd x ) = csch x ; cosh ⁡ ( cogd x ) = coth ⁡ | x | tanh ⁡ ( | cogd x | ) = sech x ; sech ( cogd x ) = tanh ⁡ | x | coth ⁡ ( | cogd x | ) = cosh ⁡ x ; csch ( cogd x ) = sinh ⁡ x {\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}}\,\!} 它們的導數分別為： d d x gd x = sech x ; d d x arcgd x = sec ⁡ x ; d d x cogd x = − csch | x | {\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}}\,\!} 在雙曲幾何中，表達式 π 2 − gd ( x ) {\displaystyle {\frac {\pi }{2}}-{\mbox{gd}}(x)} 定義了平行角（英语：Angle of parallelism）函數。 在使用麥卡托投影法的地圖，若以 y {\displaystyle y\,} 表示一個地點在地圖跟赤道的距離，則其緯度 ϕ {\displaystyle \phi \,} 和 y {\displaystyle y\,} 的關係為： ϕ = gd ( y ) {\displaystyle \phi ={\mbox{gd}}(y)\,} 古德曼函數在倒單擺的非週期解中出現。 CRC Handbook of Mathematical Sciences 5th ed. pp 323-5. Gudermannian Function -- from Wolfram MathWorld （页面存档备份，存于互联网档案馆） 克里斯托夫·古德曼（Christof Gudermann，1798年–1852年）是德國數學家，是高斯的學生，卡爾·魏爾施特拉斯的老師。[1] （页面存档备份，存于互联网档案馆）[2] （页面存档备份，存于互联网档案馆）