with real constants, has differential
|; , .|
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).