4 Elementary Functions Logarithm, Exponential, Powers 4.2 Definitions 4.4 Special Values and Limits

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

Figure 4.3.1: $\mathop{\ln\/}\nolimits x$ and $e^{x}$ .

Symbols:: : base of exponential function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and : 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 -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 -plane map onto rays in the -plane, and lines parallel to the imaginary axis in the -plane map onto circles centered at the origin in the -plane. In the labeling of corresponding points is a real parameter that can lie anywhere in the interval $(0,\infty)$ .

(i)

-plane

(ii)

-plane

A	B	C	$\overline{\mathrm{C}}$	D	$\overline{\mathrm{D}}$	E	$\overline{\mathrm{E}}$	F
0		$r+i\pi$	$r-i\pi$	$i\pi$	$-i\pi$	$-r+i\pi$	$-r-i\pi$
1	$e^{r}$	$-e^{r}+i0$	$-e^{r}-i0$			$-e^{{-r}}+i0$	$-e^{{-r}}-i0$	$e^{{-r}}$