19 Elliptic Integrals Applications 19.31 Probability Distributions 19.33 Triaxial Ellipsoids

§19.32 Conformal Map onto a Rectangle

ⓘ

Keywords:: conformal mapping, symmetric elliptic integrals
Notes:: See Carlson (1977b, pp. 234–235). For (19.32.2) use (19.18.6).
Permalink:: http://dlmf.nist.gov/19.32
See also:: Annotations for Ch.19

The function

19.32.1		$z(p)=R_{F}\left(p-x_{1},p-x_{2},p-x_{3}\right),$
ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind and $z(p)$ : function Permalink: http://dlmf.nist.gov/19.32.E1 Encodings: TeX, pMML, png See also: Annotations for §19.32 and Ch.19

with $x_{1},x_{2},x_{3}$ real constants, has differential

19.32.2		$\,\mathrm{d}z=-\frac{1}{2}\left(\prod_{j=1}^{3}(p-x_{j})^{-1/2}\right)\,% \mathrm{d}p,$
$\Im p>0$ ; $0<\operatorname{ph}\left(p-x_{j}\right)<\pi$ , $j=1,2,3$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\Im$ : imaginary part, $\operatorname{ph}$ : phase and $z(p)$ : function Referenced by: §19.32 Permalink: http://dlmf.nist.gov/19.32.E2 Encodings: TeX, pMML, png See also: Annotations for §19.32 and Ch.19

19.32.3		$x_{1}>x_{2}>x_{3},$
ⓘ Permalink: http://dlmf.nist.gov/19.32.E3 Encodings: TeX, pMML, png See also: Annotations for §19.32 and Ch.19

then $z(p)$ is a Schwartz–Christoffel mapping of the open upper-half $p$ -plane onto the interior of the rectangle in the $z$ -plane with vertices

19.32.4	$\displaystyle z(\infty)$	$\displaystyle=0,$
	$\displaystyle z(x_{1})$	$\displaystyle=R_{F}\left(0,x_{1}-x_{2},x_{1}-x_{3}\right)\quad\text{($>0$)},$
	$\displaystyle z(x_{2})$	$\displaystyle=z(x_{1})+z(x_{3}),$
	$\displaystyle z(x_{3})$	$\displaystyle=R_{F}\left(x_{3}-x_{1},x_{3}-x_{2},0\right)=-iR_{F}\left(0,x_{1}% -x_{3},x_{2}-x_{3}\right).$
ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\mathrm{i}$ : imaginary unit and $z(p)$ : function Permalink: http://dlmf.nist.gov/19.32.E4 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §19.32 and Ch.19

As $p$ proceeds along the entire real axis with the upper half-plane on the right, $z$ describes the rectangle in the clockwise direction; hence $z(x_{3})$ is negative imaginary.

For further connections between elliptic integrals and conformal maps, see Bowman (1953, pp. 44–85).

19.31 Probability Distributions 19.33 Triaxial Ellipsoids

NIST

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