§4.3 Graphics

Permalink:: http://dlmf.nist.gov/4.3

§4.3(i) Real Arguments
§4.3(ii) Complex Arguments: Conformal Maps
§4.3(iii) Complex Arguments: Surfaces

§4.3(i) Real Arguments

Notes:: This graph was produced at NIST.
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Figure 4.3.1: $\mathop{\ln\/}\nolimits x$ and $e^{x}$ .

Symbols:: $e$ : base of exponential function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $x$ : real variable
A&S Ref:: AMS55 (Figure 4.2 in simpler form)
Keywords:: exponential function, logarithm function
Permalink:: http://dlmf.nist.gov/4.3.F1
Encodings:: pdf, png

§4.3(ii) Complex Arguments: Conformal Maps

Notes:: This conformal map was produced at NIST.
Keywords:: exponential function, logarithm function
Permalink:: http://dlmf.nist.gov/4.3.ii

Figure 4.3.2 illustrates the conformal mapping of the strip $-\pi<\imagpart{z}<\pi$ onto the whole $w$ -plane cut along the negative real axis, where $w=e^{z}$ and $z=\mathop{\ln\/}\nolimits w$ (principal value). Corresponding points share the same letters, with bars signifying complex conjugates. Lines parallel to the real axis in the $z$ -plane map onto rays in the $w$ -plane, and lines parallel to the imaginary axis in the $z$ -plane map onto circles centered at the origin in the $w$ -plane. In the labeling of corresponding points $r$ is a real parameter that can lie anywhere in the interval $(0,\infty)$ .

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