# §19.32 Conformal Map onto a Rectangle

The function

with real constants, has differential

19.32.2; , .

If

19.32.3

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

As proceeds along the entire real axis with the upper half-plane on the right, describes the rectangle in the clockwise direction; hence is negative imaginary.

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