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19 Elliptic IntegralsApplications

§19.32 Conformal Map onto a Rectangle

The function

19.32.1 z(p)=RF(p-x1,p-x2,p-x3),

with x1,x2,x3 real constants, has differential

19.32.2 dz=-12(j=13(p-xj)-1/2)dp,
p>0; 0<ph(p-xj)<π, j=1,2,3.


19.32.3 x1>x2>x3,

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() =0,
z(x1) =RF(0,x1-x2,x1-x3)(>0),
z(x2) =z(x1)+z(x3),
z(x3) =RF(x3-x1,x3-x2,0)=-iRF(0,x1-x3,x2-x3).

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(x3) is negative imaginary.

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