# §19.32 Conformal Map onto a Rectangle

The function

 19.32.1 $z(p)=\mathop{R_{F}\/}\nolimits\!\left(p-x_{1},p-x_{2},p-x_{3}\right),$ Symbols: $\mathop{R_{F}\/}\nolimits\!\left(\NVar{x},\NVar{y},\NVar{z}\right)$: 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

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

 19.32.2 $\mathrm{d}z=-\frac{1}{2}\left(\prod_{j=1}^{3}(p-x_{j})^{-1/2}\right)\mathrm{d}p,$ $\Im{p}>0$; $0<\mathop{\mathrm{ph}\/}\nolimits\!\left(p-x_{j}\right)<\pi$, $j=1,2,3$.

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

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

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).