A method for conformal mapping of a two-connected region onto an annulus

Abstract

This paper presents a method for conformal of a two-connected region k i onto an annulus.The philosophy of the method is to convert the p ~oblem into a Dirichlet problem and to prove the real part of the analytic" function transformation should ~ a harmonic function satisfbing certain boundary conditions. According tc the theo,-'y of hazmonic function we can determine t)e inner radius of the annulus from the condition that the harmonic function uefined in two-connected region should be slngle-valued. It is then easy to see that the, imaginary palt can directly be obtained with the aid of Cauchy-Riemann conditions.The. unknown constants of integration only influence the argumen~ of fmag(~ poii.ts and can easily be derived bq using the one-to-one mapping of ~ region onto an annulus. Without loss of general:ty, t:'~. method can be used to conformally map other two-connected regions onto an annulus if they can be subdivided into sevoral zeccangulars.The method has been programmed for a digital computer. It is demonstrated that the method is efficient and economical. The corresponding numerical results are shown in the Table. ?

Torsion of tubes with quasi-polygonal holes using complex variable method

“…If σ = e iθ , from Harnack’s theorem [27] we can finally arrive at…”