Conformal mapping of an $L$-shaped domain in analytical form

The problem of finding parameters of the Schwarz–Christoffel integral for a conformal mapping $f$ of a canonical domain onto an $L$-shaped one is solved analytically for arbitrary geometric parameters of the domain. The unknown preimage is represented in the form of a series in powers of a small parameter with coefficients written in closed form, and an estimate for the moduli of the coefficients is obtained. We find asymptotics for the crowding effect (crowding of preimages), which is especially pronounced for elongated domains are computing The mapping $f$ and its inverse $f^{-1}$ are computed using series with closed-form coefficients, whose domains of convergence collectively cover the entire (closed) mapped domain. Combining $f$ with linear fractional mappings and the elliptic sine function yields mappings of the half-plane, disk, and rectangle onto an $L$-shaped domain. Numerical implementations of the constructed mappings demonstrate the high efficiency of the applied methods.