inverse function

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11: 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 $(\sinh\phi)R_{C}\left(1,{\cosh}^{2}\phi\right)=\operatorname{gd}\left(\phi% \right).$
12: 4.15 Graphics
§4.15(iii) Complex Arguments: Surfaces
The corresponding surfaces for $\operatorname{arccos}\left(x+iy\right)$, $\operatorname{arccot}\left(x+iy\right)$, $\operatorname{arcsec}\left(x+iy\right)$ can be visualized from Figures 4.15.9, 4.15.11, 4.15.13 with the aid of equations (4.23.16)–(4.23.18).
13: 27.5 Inversion Formulas
§27.5 Inversion Formulas
27.5.2 $\sum_{d\mathbin{|}n}\mu\left(d\right)=\left\lfloor\frac{1}{n}\right\rfloor,$
27.5.3 $g(n)=\sum_{d\mathbin{|}n}f(d)\Longleftrightarrow f(n)=\sum_{d\mathbin{|}n}g(d)% \mu\left(\frac{n}{d}\right).$
14: 4.32 Inequalities
4.32.4 $\operatorname{arctan}x\leq\tfrac{1}{2}\pi\tanh x,$ $x\geq 0$.
16: 4.39 Continued Fractions
§4.39 Continued Fractions
4.39.2 $\frac{\operatorname{arcsinh}z}{\sqrt{1+z^{2}}}=\cfrac{z}{1+\cfrac{1\cdot 2z^{2% }}{3+\cfrac{1\cdot 2z^{2}}{5+\cfrac{3\cdot 4z^{2}}{7+\cfrac{3\cdot 4z^{2}}{9+% \cdots}}}}},$
4.39.3 $\operatorname{arctanh}z=\cfrac{z}{1-\cfrac{z^{2}}{3-\cfrac{4z^{2}}{5-\cfrac{9z% ^{2}}{7-\cdots}}}},$
For these and other continued fractions involving inverse hyperbolic functions see Lorentzen and Waadeland (1992, pp. 569–571). …
18: 8.12 Uniform Asymptotic Expansions for Large Parameter
InverseFunction
For asymptotic expansions, as $a\to\infty$, of the inverse function $x=x(a,q)$ that satisfies the equation …These expansions involve the inverse error function $\operatorname{inverfc}\left(x\right)$7.17), and are uniform with respect to $q\in[0,1]$. …
19: 4.24 Inverse Trigonometric Functions: Further Properties
§4.24(ii) Derivatives
4.24.17 $\operatorname{Arctan}u\pm\operatorname{Arccot}v=\operatorname{Arctan}\left(% \frac{uv\pm 1}{v\mp u}\right)=\operatorname{Arccot}\left(\frac{v\mp u}{uv\pm 1% }\right).$
4.38.19 $\operatorname{Arctanh}u\pm\operatorname{Arccoth}v=\operatorname{Arctanh}\left(% \frac{uv\pm 1}{v\pm u}\right)=\operatorname{Arccoth}\left(\frac{v\pm u}{uv\pm 1% }\right).$