Walsh’s Conformal Map onto Lemniscatic Domains for Several Intervals

We consider Walsh’s conformal map from the complement of a compact set E=∪j=1ℓEj with ℓ components onto a lemniscatic domain C^\L, where L has the form L={w∈C:∏j=1ℓ|w-aj|mj≤cap(E)}. We prove that the exponents mj appearing in L satisfy mj=μE(Ej), where μE is the equilibrium measure of E. When E is the union of ℓ real intervals, we derive a fast algorithm for computing the centers a1,…,aℓ. For ℓ=2, the formulas for m1,m2 and a1,a2 are explicit. Moreover, we obtain the conformal map numerically. Our approach relies on the real and complex Green’s functions of C^\E and C^\L.