On conformal mapping of a multiply-connected domain onto a circular slit covering surface
Hisao Mizumoto · Kodai Mathematical Journal · 1961
In the present paper we will concern ourselves with conformal mapping of a multiply-connected domain of finite connectivity onto a canonical covering surface whose boundary consists of whole circumferences and circular slits centred at the origin on the basic plane.We will discuss the existence of such a mapping function and its extremality.The purpose of our present investigation is an extension and an improvement of the results obtained in our previous papers [4] and [5].§ 2. Preliminaries.Let B be a multiply-connected domain of finite connectivity on the z-plane.We suppose that each component C 3 (j = l, ,N) of its boundary C is a continuum.Let z 0 ,z°k (fc = l, ,JV°; Λf°^0) and zT (1 = 1, , N°°; ΛΓ ro ^O) be arbitrarily preassigned N° + N°° + l points in B, and positive integers μl and μT (fe = 1, , ΛΓ°; I = 1, , jV°°) be given arbitrarily.