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4 Elementary Functions

4.2.1 Branch cut for $\ln z$ and $z^{\alpha}$ .
4.3.1 $\ln x$ and $e^{x}$ .
(i) $z$ -plane (ii) $w$ -plane

A B C $\overline{\mathrm{C}}$ D $\overline{\mathrm{D}}$ E $\overline{\mathrm{E}}$ F

$z$ 0 $r$ $r+i\pi$ $r-i\pi$ $i\pi$ $-i\pi$ $-r+i\pi$ $-r-i\pi$ $-r$

$w$ 1 $e^{r}$ $-e^{r}+i0$ $-e^{r}-i0$ $-1+i0$ $-1-i0$ $-e^{-r}+i0$ $-e^{-r}-i0$ $e^{-r}$

4.3.2 Conformal mapping of exponential and logarithm.
4.3.3 $\ln\left(x+iy\right)$ .
4.3.4 $e^{x+iy}$ .
4.13.1 Branches $\mathrm{Wp}\left(x\right)$ , $\mathrm{Wm}\left(x\right)$ of the Lambert $W$ -function.
4.15.1 $\sin x$ and $\cos x$ .
4.15.2 $\operatorname{Arcsin}x$ and $\operatorname{Arccos}x$ .
4.15.3 $\tan x$ and $\cot x$ .
4.15.4 $\operatorname{arctan}x$ and $\operatorname{arccot}x$ .
4.15.5 $\csc x$ and $\sec x$ .
4.15.6 $\operatorname{arccsc}x$ and $\operatorname{arcsec}x$ .

See accompanying text

4.15.7 Conformal mapping of sine and inverse sine.

4.15.8 $\sin\left(x+iy\right)$ .
4.15.9 $\operatorname{arcsin}\left(x+iy\right)$ .
4.15.10 $\tan\left(x+iy\right)$ .
4.15.11 $\operatorname{arctan}\left(x+iy\right)$ .
4.15.12 $\csc\left(x+iy\right)$ .
4.15.13 $\operatorname{arccsc}\left(x+iy\right)$ .
4.16.1 Quadrants for the angle $\theta$ .
(i) $\operatorname{arcsin}z$ and $\operatorname{arccos}z$ (ii) $\operatorname{arctan}z$ (iii) $\operatorname{arccsc}z$ and $\operatorname{arcsec}z$ (iv) $\operatorname{arccot}z$

4.23.1 Branch cuts for the inverse trigonometric functions.
4.29.1 $\sinh x$ and $\cosh x$ .
4.29.2 $\operatorname{arcsinh}x$ and $\operatorname{arccosh}x$ .
4.29.3 $\tanh x$ and $\coth x$ .
4.29.4 $\operatorname{arctanh}x$ and $\operatorname{arccoth}x$ .
4.29.5 $\operatorname{csch}x$ and $\operatorname{sech}x$ .
4.29.6 $\operatorname{arccsch}x$ and $\operatorname{arcsech}x$ .
(i) $\operatorname{arcsinh}z$ (ii) $\operatorname{arccosh}z$ (iii) $\operatorname{arctanh}z$

(iv) $\operatorname{arccsch}z$ (v) $\operatorname{arcsech}z$ (vi) $\operatorname{arccoth}z$

4.37.1 Branch cuts for the inverse hyperbolic functions.
4.42.1 Planar right triangle.
4.42.2 Planar triangle.
4.42.3 Spherical triangle.

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