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This paper presents a new uniquely solvable boundary integral equation forcomputing the conformal mapping, its derivative and its inverse from boundedmultiply connected regions onto the five classical canonical slit regions. Theintegral equation is derived by reformulating the conformal mapping as anadjoint Riemann-Hilbert problem. From the adjoint Riemann-Hilbert problem, wederive a boundary integral equation with the adjoint generalized Neumann kernelfor the derivative of the boundary correspondence function $\theta'$. Only theright-hand side of the integral equation is different from a canonical regionto another. The function $\theta'$ is integrated to obtain the boundarycorrespondence function $\theta$. The integration constants as well as theparameters of the canonical region are computed using the same uniquelysolvable integral equation.  A numerical example is presented to illustrate the accuracy of the proposedmethod.

Numerical conformal mapping via a boundary integral equation with the adjoint generalized Neumann kernel