Zolotarev's conformal mapping and Chebotarev's problem

Abstract

Consider the Chebotarev problem of finding a continuum S in the complex plane including some given points such that the logarithmic capacity of S is minimal. In this paper, we give a complete solution of this problem for the case of three given points with the help of Zolotarev's conformal mapping using Jacobian elliptic and theta functions. Moreover, for four given points, some special cases can be treated.

Keywords:

• Chebotarev's problem
• Jacobian elliptic function
• Jacobian theta function
• logarithmiccapacity

Mathematics Subject Classification:

• 30C20
• 30C85
• 33E05
• 31A15