تربط الدالة الغودرمانية[1] أو دالة غودرمان، التي سميت على اسم كريستوف غودرمان (1798–1852)، الدوال المثلثية بالدوال الزائدية دون استخدام الأعداد المركبة.
تعرّف بـ :[2][3][4]
![{\displaystyle {\begin{aligned}\operatorname {gd} x&=\arcsin \left(\tanh x\right)=\arctan(\sinh x)=\operatorname {arccsc} (\coth x)\\[6pt]&=\operatorname {sgn}(x)\arccos \left(\operatorname {sech} x\right)=\operatorname {sgn}(x)\operatorname {arcsec} (\cosh x)\\[6pt]&=2\arctan \left(\tanh {\frac {x}{2}}\right)\\[6pt]&=2\arctan \left(e^{x}\right)-{\frac {\pi }{2}}=-i\ln \left(\operatorname {sech} x+i\tanh x\right)\\[6pt]&=-{\frac {i}{2}}\ln \left({\frac {1+i\sinh x}{1-i\sinh x}}\right)=-i\ln \left({\frac {1+i\sinh x}{\cosh x}}\right)\\[6pt]&=-i\ln \left({\frac {1+i\tanh {\frac {x}{2}}}{1-i\tanh {\frac {x}{2}}}}\right)=-i\ln \left(i\tanh \left({\frac {x}{2}}-{\frac {i\pi }{4}}\right)\right)~.\end{aligned}}}](./a19def014c60dee0218cf428f91e2ff5e9e67aac.svg)
![{\displaystyle {\begin{aligned}\sin(\operatorname {gd} x)&=\tanh x\;;&\csc(\operatorname {gd} x)&=\coth x\;;\\[6pt]\cos(\operatorname {gd} x)&=\operatorname {sech} x\;;&\sec(\operatorname {gd} x)&=\cosh x\;;\\[6pt]\tan(\operatorname {gd} x)&=\sinh x\;;&\cot(\operatorname {gd} x)&=\operatorname {csch} x\;;\\[6pt]\tan \left({\frac {\operatorname {gd} x}{2}}\right)&=\tanh {\frac {x}{2}}~.\end{aligned}}}](./d39ac08c63e88ee283040567fca2f502ff2f4baf.svg)
رسم بياني لمعكوس دالة غودرمانية
![{\displaystyle {\begin{aligned}\operatorname {gd} ^{-1}x&=\int _{0}^{x}\sec t\,\mathrm {d} t\qquad {\text{ for }}\quad -{\frac {\pi }{2}}<x<{\frac {\pi }{2}}\\[8pt]&=\ln \left|{\frac {1+\sin x}{\cos x}}\right|={\frac {1}{2}}\ln \left|{\frac {1+\sin x}{1-\sin x}}\right|=\ln \left|{\frac {1+\tan {\frac {x}{2}}}{1-\tan {\frac {x}{2}}}}\right|\\[8pt]&=\ln \left|\tan x+\sec x\right|=\ln \left|\tan \left({\frac {x}{2}}+{\frac {\pi }{4}}\right)\right|\\[8pt]&=\operatorname {artanh} (\sin x)=\operatorname {arsinh} (\tan x)\\[6pt]&=2\operatorname {artanh} \left(\tan {\frac {x}{2}}\right)\\[6pt]&=\operatorname {arcoth} (\csc x)=\operatorname {arcsch} (\cot x)\\[6pt]&=\operatorname {sgn} (x)\operatorname {arcosh} (\sec x)=\operatorname {sgn} (x)\operatorname {arsech} (\cos x)\\[6pt]&=-i\operatorname {gd} (ix)~.\end{aligned}}}](./a57e462caa649cc054ce0fee8f724c207b375c62.svg)
![{\displaystyle {\begin{aligned}\sinh \left(\operatorname {gd} ^{-1}x\right)&=\tan x\;;&\operatorname {csch} \left(\operatorname {gd} ^{-1}x\right)&=\cot x\;;\\[6pt]\cosh \left(\operatorname {gd} ^{-1}x\right)&=\sec x\;;&\operatorname {sech} \left(\operatorname {gd} ^{-1}x\right)&=\cos x\;;\\[6pt]\tanh \left(\operatorname {gd} ^{-1}x\right)&=\sin x\;;&\coth \left(\operatorname {gd} ^{-1}x\right)&=\csc x~.\end{aligned}}}](./4158261eefcd01f5d33b8153415a345dec3afb59.svg)
![{\displaystyle {\begin{aligned}\operatorname {gd} ^{-1}x=\int _{0}^{x}\sec t\,\mathrm {d} t\\[6pt]=\ln \left|\tan x+\sec x\right|\end{aligned}}}](./66f7c111a69ae46557ed6c824611a7ad424e2994.svg)
![{\displaystyle \left]-{\frac {\pi }{2}},{\frac {\pi }{2}}\right[}](./c6b123273d134ae0a355186bca3947bf36996ceb.svg)
