§19.32 Conformal Map onto a Rectangle

The function

19.32.1		$$z(p)=R_F(p-x_1,p-x_2,p-x_3),$$
ⓘ Symbols: $R_F(x,y,z)$: 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		$$dz=-\frac{1}{2}\left(\prod _{j=1}^3{(p-x_j)}^{-1/2}\right)dp,$$
$\mathrm{\Im }p>0$; , $j=1,2,3$.
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{\Im }$: imaginary part, $\mathrm{ph}$: 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

If

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		$z(\mathrm{\infty })$	$=0,$
		$z(x_1)$	$=R_F(0,x_1-x_2,x_1-x_3)\hspace{1em}\text{(}>0\text{)},$
		$z(x_2)$	$=z(x_1)+z(x_3),$
		$z(x_3)$	$=R_F(x_3-x_1,x_3-x_2,0)=-\mathrm{i}{R}_F(0,x_1-x_3,x_2-x_3).$
ⓘ Symbols: $R_F(x,y,z)$: 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).