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

§4.3 Graphics

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Permalink:: http://dlmf.nist.gov/4.3
See also:: Annotations for Ch.4

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

§4.3(i) Real Arguments

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Notes:: This graph was produced at NIST.
Permalink:: http://dlmf.nist.gov/4.3.i
See also:: Annotations for §4.3 and Ch.4

See accompanying text — Figure 4.3.1: $\ln x$ and $e^{x}$ . Parallel tangent lines at $(1,0)$ and $(0,1)$ make evident the mirror symmetry across the line $y=x$ , demonstrating the inverse relationship between the two functions. Magnify

§4.3(ii) Complex Arguments: Conformal Maps

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Keywords:: conformal maps, exponential function, logarithm function
Notes:: This conformal map was produced at NIST.
Permalink:: http://dlmf.nist.gov/4.3.ii
See also:: Annotations for §4.3 and Ch.4

Figure 4.3.2 illustrates the conformal mapping of the strip $-\pi<\Im z<\pi$ onto the whole $w$ -plane cut along the negative real axis, where $w=e^{z}$ and $z=\ln w$ (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 $(0,\infty)$ .

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