# inversion

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##### 1: 7.17 Inverse Error Functions
###### §7.17(i) Notation
The inverses of the functions $x=\operatorname{erf}y$, $x=\operatorname{erfc}y$, $y\in\mathbb{R}$, are denoted by …
##### 2: 4.37 Inverse Hyperbolic Functions
###### §4.37 Inverse Hyperbolic Functions
4.37.4 $\operatorname{Arccsch}z=\operatorname{Arcsinh}\left(1/z\right),$
4.37.7 $\operatorname{arccsch}z=\operatorname{arcsinh}\left(1/z\right),$
4.37.8 $\operatorname{arcsech}z=\operatorname{arccosh}\left(1/z\right).$
4.37.9 $\operatorname{arccoth}z=\operatorname{arctanh}\left(1/z\right),$ $z\neq\pm 1$.
##### 3: 22.15 Inverse Functions
###### §22.15(i) Definitions
Each of these inverse functions is multivalued. The principal values satisfy …
##### 4: 4.23 Inverse Trigonometric Functions
###### §4.23 Inverse Trigonometric Functions
4.23.6 $\operatorname{Arccot}z=\operatorname{Arctan}\left(1/z\right).$
4.23.7 $\operatorname{arccsc}z=\operatorname{arcsin}\left(1/z\right),$
4.23.8 $\operatorname{arcsec}z=\operatorname{arccos}\left(1/z\right).$
4.23.9 $\operatorname{arccot}z=\operatorname{arctan}\left(1/z\right),$ $z\neq\pm\mathrm{i}$.
18.28.10 $\sum_{y=0}^{\infty}\frac{(1-q^{2y}a^{-2})\left(a^{-2},(ab)^{-1};q\right)_{y}}{% (1-a^{-2})\left(q,bqa^{-1};q\right)_{y}}(ba^{-1})^{y}q^{y^{2}}\*Q_{n}\left(% \tfrac{1}{2}(aq^{-y}+a^{-1}q^{y});a,b\,|\,q^{-1}\right)\*Q_{m}\left(\tfrac{1}{% 2}(aq^{-y}+a^{-1}q^{y});a,b\,|\,q^{-1}\right)=\frac{\left(qa^{-2};q\right)_{% \infty}}{\left(ba^{-1}q;q\right)_{\infty}}\left(q,(ab)^{-1};q\right)_{n}(ab)^{% n}q^{-n^{2}}\delta_{n,m}.$
18.28.18 $h_{n}\left(\sinh t\,|\,q\right)=\sum_{\ell=0}^{n}q^{\frac{1}{2}\ell(\ell+1)}% \frac{\left(q^{-n};q\right)_{\ell}}{\left(q;q\right)_{\ell}}e^{(n-2\ell)t}=e^{% nt}{{}_{1}\phi_{1}}\left({q^{-n}\atop 0};q,-qe^{-2t}\right)={\mathrm{i}^{-n}}H% _{n}\left(\mathrm{i}\sinh t\,|\,q^{-1}\right).$
##### 6: 19.2 Definitions
19.2.17 $R_{C}\left(x,y\right)=\frac{1}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t+x% }(t+y)},$
In (19.2.18)–(19.2.22) the inverse trigonometric and hyperbolic functions assume their principal values (§§4.23(ii) and 4.37(ii)). When $x$ and $y$ are positive, $R_{C}\left(x,y\right)$ is an inverse circular function if $x and an inverse hyperbolic function (or logarithm) if $x>y$:
19.2.18 $R_{C}\left(x,y\right)=\frac{1}{\sqrt{y-x}}\operatorname{arctan}\sqrt{\frac{y-x% }{x}}=\frac{1}{\sqrt{y-x}}\operatorname{arccos}\sqrt{x/y},$ $0\leq x,
19.2.19 $R_{C}\left(x,y\right)=\frac{1}{\sqrt{x-y}}\operatorname{arctanh}\sqrt{\frac{x-% y}{x}}=\frac{1}{\sqrt{x-y}}\ln\frac{\sqrt{x}+\sqrt{x-y}}{\sqrt{y}},$ $0.
##### 7: 4.27 Sums
###### §4.27 Sums
For sums of trigonometric and inverse trigonometric functions see Gradshteyn and Ryzhik (2000, Chapter 1), Hansen (1975, §§14–42), Oberhettinger (1973), and Prudnikov et al. (1986a, Chapter 5).
##### 8: 4.47 Approximations
###### §4.47(i) Chebyshev-Series Expansions
Clenshaw (1962) and Luke (1975, Chapter 3) give 20D coefficients for $\ln$, $\exp$, $\sin$, $\cos$, $\tan$, $\cot$, $\operatorname{arcsin}$, $\operatorname{arctan}$, $\operatorname{arcsinh}$. … Hart et al. (1968) give $\ln$, $\exp$, $\sin$, $\cos$, $\tan$, $\cot$, $\operatorname{arcsin}$, $\operatorname{arccos}$, $\operatorname{arctan}$, $\sinh$, $\cosh$, $\tanh$, $\operatorname{arcsinh}$, $\operatorname{arccosh}$. … Luke (1975, Chapter 3) supplies real and complex approximations for $\ln$, $\exp$, $\sin$, $\cos$, $\tan$, $\operatorname{arctan}$, $\operatorname{arcsinh}$. …
##### 9: 4.29 Graphics
###### §4.29(ii) Complex Arguments
The surfaces for the complex hyperbolic and inverse hyperbolic functions are similar to the surfaces depicted in §4.15(iii) for the trigonometric and inverse trigonometric functions. …
##### 10: 4.1 Special Notation
The main purpose of the present chapter is to extend these definitions and properties to complex arguments $z$. The main functions treated in this chapter are the logarithm $\ln z$, $\operatorname{Ln}z$; the exponential $\exp z$, $e^{z}$; the circular trigonometric (or just trigonometric) functions $\sin z$, $\cos z$, $\tan z$, $\csc z$, $\sec z$, $\cot z$; the inverse trigonometric functions $\operatorname{arcsin}z$, $\operatorname{Arcsin}z$, etc. ; the hyperbolic trigonometric (or just hyperbolic) functions $\sinh z$, $\cosh z$, $\tanh z$, $\operatorname{csch}z$, $\operatorname{sech}z$, $\coth z$; the inverse hyperbolic functions $\operatorname{arcsinh}z$, $\operatorname{Arcsinh}z$, etc. Sometimes in the literature the meanings of $\ln$ and $\operatorname{Ln}$ are interchanged; similarly for $\operatorname{arcsin}z$ and $\operatorname{Arcsin}z$, etc. …${\sin^{-1}}z$ for $\operatorname{arcsin}z$ and $\mathrm{Sin}^{-1}\;z$ for $\operatorname{Arcsin}z$.