# inverse Gudermannian function

(0.003 seconds)

## 8 matching pages

##### 1: 19.6 Special Cases
19.6.8 $F\left(\phi,1\right)=(\sin\phi)R_{C}\left(1,{\cos}^{2}\phi\right)={% \operatorname{gd}^{-1}}\left(\phi\right).$
For the inverse Gudermannian function ${\operatorname{gd}^{-1}}\left(\phi\right)$ see §4.23(viii). …
##### 2: 4.23 Inverse Trigonometric Functions
4.23.40 $\operatorname{gd}\left(x\right)=2\operatorname{arctan}\left(e^{x}\right)-% \tfrac{1}{2}\pi\\ =\operatorname{arcsin}\left(\tanh x\right)=\operatorname{arccsc}\left(\coth x% \right)\\ =\operatorname{arccos}\left(\operatorname{sech}x\right)=\operatorname{arcsec}% \left(\cosh x\right)\\ =\operatorname{arctan}\left(\sinh x\right)=\operatorname{arccot}\left(% \operatorname{csch}x\right).$
The inverse Gudermannian function is given by
4.23.41 ${\operatorname{gd}^{-1}}\left(x\right)=\int_{0}^{x}\sec t\,\mathrm{d}t,$ $-\frac{1}{2}\pi.
4.23.42 ${\operatorname{gd}^{-1}}\left(x\right)=\ln\tan\left(\tfrac{1}{2}x+\tfrac{1}{4}% \pi\right)=\ln\left(\sec x+\tan x\right)=\operatorname{arcsinh}\left(\tan x% \right)=\operatorname{arccsch}\left(\cot x\right)=\operatorname{arccosh}\left(% \sec x\right)=\operatorname{arcsech}\left(\cos x\right)=\operatorname{arctanh}% \left(\sin x\right)=\operatorname{arccoth}\left(\csc x\right).$
##### 4: 19.10 Relations to Other Functions
For relations to the Gudermannian function $\operatorname{gd}\left(x\right)$ and its inverse ${\operatorname{gd}^{-1}}\left(x\right)$4.23(viii)), see (19.6.8) and
19.10.2 $(\sinh\phi)R_{C}\left(1,{\cosh}^{2}\phi\right)=\operatorname{gd}\left(\phi% \right).$
##### 5: 4.26 Integrals
4.26.5 $\int\sec x\,\mathrm{d}x={\operatorname{gd}^{-1}}\left(x\right),$ $-\frac{1}{2}\pi.
##### 6: 19.9 Inequalities
19.9.11 $\phi\leq F\left(\phi,k\right)\leq\min(\phi/\Delta,{\operatorname{gd}^{-1}}% \left(\phi\right)),$
##### 7: 22.16 Related Functions
###### §22.16(i) Jacobi’s Amplitude ($\operatorname{am}$) Function
where the inverse sine has its principal value when $-K\leq x\leq K$ and is defined by continuity elsewhere. … For the Gudermannian function $\operatorname{gd}\left(x\right)$ see §4.23(viii). …
##### 8: 4.40 Integrals
###### §4.40(iv) Inverse Hyperbolic Functions
Extensive compendia of indefinite and definite integrals of hyperbolic functions include Apelblat (1983, pp. 96–109), Bierens de Haan (1939), Gröbner and Hofreiter (1949, pp. 139–160), Gröbner and Hofreiter (1950, pp. 160–167), Gradshteyn and Ryzhik (2000, Chapters 2–4), and Prudnikov et al. (1986a, §§1.4, 1.8, 2.4, 2.8).