Gudermannian function

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2: 19.10 Relations to Other Functions
In each case when $y=1$, the quantity multiplying $R_{C}$ supplies the asymptotic behavior of the left-hand side as the left-hand side tends to 0. 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 $(\sinh\phi)R_{C}\left(1,{\cosh}^{2}\phi\right)=\operatorname{gd}\left(\phi% \right).$
3: 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). …
4: 4.23 Inverse Trigonometric Functions
§4.23(viii) GudermannianFunction
4.23.39 $\operatorname{gd}\left(x\right)=\int_{0}^{x}\operatorname{sech}t\,\mathrm{d}t,$ $-\infty.
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 ${\operatorname{gd}^{-1}}\left(x\right)=\ln\tan\left(\tfrac{1}{2}x+\tfrac{1}{4}% \pi\right)=\ln\left(\sec x+\tan x\right)=\operatorname{arcsinh}\left(\tan x% \right)=\operatorname{arccsch}\left(\cot x\right)=\operatorname{arccosh}\left(% \sec x\right)=\operatorname{arcsech}\left(\cos x\right)=\operatorname{arctanh}% \left(\sin x\right)=\operatorname{arccoth}\left(\csc x\right).$
5: 22.16 Related Functions
22.16.4 $\operatorname{am}\left(x,0\right)=x,$
For the Gudermannian function $\operatorname{gd}\left(x\right)$ see §4.23(viii). …
22.16.8 $\operatorname{am}\left(x,k\right)=\operatorname{gd}x-\tfrac{1}{4}{k^{\prime}}^% {2}(x-\sinh x\cosh x)\operatorname{sech}x+O\left({k^{\prime}}^{4}\right).$
7: 4.26 Integrals
4.26.5 $\int\sec x\,\mathrm{d}x={\operatorname{gd}^{-1}}\left(x\right),$ $-\frac{1}{2}\pi.
8: 19.9 Inequalities
19.9.11 $\phi\leq F\left(\phi,k\right)\leq\min(\phi/\Delta,{\operatorname{gd}^{-1}}% \left(\phi\right)),$
9: 22.1 Special Notation
The notation $\operatorname{sn}\left(z,k\right)$, $\operatorname{cn}\left(z,k\right)$, $\operatorname{dn}\left(z,k\right)$ is due to Gudermann (1838), following Jacobi (1827); that for the subsidiary functions is due to Glaisher (1882). …