19 Elliptic Integrals Applications 19.31 Probability Distributions 19.33 Triaxial Ellipsoids

§19.32 Conformal Map onto a Rectangle

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

The function

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

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind and : function
Permalink:: http://dlmf.nist.gov/19.32.E1
Encodings:: TeX, pMML, png

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

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

Symbols:: : differential of , $\imagpart{}$ : imaginary part, $\mathop{\mathrm{ph}\/}\nolimits$ : phase and : function
Referenced by:: §19.32
Permalink:: http://dlmf.nist.gov/19.32.E2
Encodings:: TeX, pMML, png

19.32.3 $x_{1}>x_{2}>x_{3},$

Permalink:: http://dlmf.nist.gov/19.32.E3
Encodings:: TeX, pMML, png

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

19.32.4

$z(\infty)=0,$

$z(x_{1})=\mathop{R_{F}\/}\nolimits\!\left(0,x_{1}-x_{2},x_{1}-x_{3}\right)% \quad\text{($>0$)},$

$z(x_{2})=z(x_{1})+z(x_{3}),$

$z(x_{3})=\mathop{R_{F}\/}\nolimits\!\left(x_{3}-x_{1},x_{3}-x_{2},0\right)=-i% \mathop{R_{F}\/}\nolimits\!\left(0,x_{1}-x_{3},x_{2}-x_{3}\right).$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind and : function
Permalink:: http://dlmf.nist.gov/19.32.E4
Encodings:: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png

As proceeds along the entire real axis with the upper half-plane on the right, 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).

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 19.31 Probability Distributions 19.33 Triaxial Ellipsoids