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

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov

§19.32 Conformal Map onto a Rectangle

pFad - (p)hone/(F)rame/(a)nonymizer/(d)eclutterfier! Saves Data!