Gudermannova funkcija
Gudermannianova funkcija, nazvana po Christophu Gudermannu (1798. – 1852. godina), povezuje kružne i hiperboličke funkcije bez korištenja kompleksnih brojeva. Definisana je sa g d ( x ) = ∫ 0 x d p cosh ⁡ ( p ) = arcsin ⁡ ( tanh ⁡ ( x ) ) = arccsc ( coth ⁡ ( x ) ) = arccos ⁡ ( sech ( x ) ) = arcsec ( cosh ⁡ ( x ) ) = arctan ⁡ ( sinh ⁡ ( x ) ) = arccot ( csch ( x ) ) = 2 arctan ⁡ ( tanh ⁡ ( x 2 ) ) = 2 arctan ⁡ ( e x ) − π 2 . {\displaystyle {\begin{aligned}{\rm {gd}}(x)&=\int _{0}^{x}{\frac {dp}{\cosh(p)}}\\&=\arcsin \left(\tanh(x)\right)={\mbox{arccsc}}\left(\coth(x)\right)\\&=\arccos \left({\mbox{sech}}(x)\right)={\mbox{arcsec}}\left(\cosh(x)\right)\\&=\arctan \left(\sinh(x)\right)={\mbox{arccot}}\left({\mbox{csch}}(x)\right)\\&=2\arctan \left(\tanh \left({\frac {x}{2}}\right)\right)=2\arctan(e^{x})-{\frac {\pi }{2}}.\end{aligned}}\,\!} Sljedeći identiteti, također, važe: sin ⁡ ( gd ( x ) ) ˙ = tanh ⁡ ( x ) ; csc ⁡ ( gd ( x ) ) = coth ⁡ ( x ) ; cos ⁡ ( gd ( x ) ) = sech ( x ) ; sec ⁡ ( gd ( x ) ) = cosh ⁡ ( x ) ; tan ⁡ ( gd ( x ) ) = sinh ⁡ ( x ) ; cot ⁡ ( gd ( x ) ) = csch ( x ) ; . tan ⁡ ( gd ( x ) 2 ) = tanh ⁡ ( x 2 ) . {\displaystyle {\begin{aligned}{\color {white}{\dot {\color {black}\sin({\mbox{gd}}(x))}}}&=\tanh(x);\quad \csc({\mbox{gd}}(x))=\coth(x);\\\cos({\mbox{gd}}(x))&={\mbox{sech}}(x);\quad \,\sec({\mbox{gd}}(x))=\cosh(x);\\\tan({\mbox{gd}}(x))&=\sinh(x);\quad \,\cot({\mbox{gd}}(x))={\mbox{csch}}(x);\\{}_{\color {white}.}\tan \left({\frac {{\mbox{gd}}(x)}{2}}\right)&=\tanh \left({\frac {x}{2}}\right).\end{aligned}}\,\!} Inverzna Gudermannova funkcija je data sa arcgd ( x ) = g d − 1 ( x ) = ∫ 0 x d p cos ⁡ ( p ) = arccosh ( sec ⁡ ( x ) ) = arctanh ( sin ⁡ ( x ) ) = 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 {dp}{\cos(p)}}\\&={}{\mbox{arccosh}}(\sec(x))={\mbox{arctanh}}(\sin(x))\\&={}\ln \left(\sec(x)(1+\sin(x))\right)\\&={}\ln(\tan(x)+\sec(x))=\ln \tan \left({\frac {\pi }{4}}+{\frac {x}{2}}\right)\\&={}{\frac {1}{2}}\ln {\frac {1+\sin(x)}{1-\sin(x)}}.\end{aligned}}\,\!} Derivacije Gudermannove funkcije i njenih inverzna su d d x gd ( x ) = sech ( x ) ; d d x arcgd ( x ) = sec ⁡ ( x ) . {\displaystyle {\frac {d}{dx}}\;{\mbox{gd}}(x)={\mbox{sech}}(x);\quad {\frac {d}{dx}}\;{\mbox{arcgd}}(x)=\sec(x).\,\!}