Collective singularities of families of analytic functions

[摘要] In his thesis Johnson gives the following definition of the radius of regularity of a family of holomorphic functions: Let F denote a family of functions f(z) regular at Z0. R is called the radius of regularity of F at z0 if R is the largest number r such that each function is holomorphic and the family is normal in |z - z 0| < r. If the conditions are valid in |z - z0| < r for each r > 0, then R = infinity. If the conditions are not valid in |z - z0| < r for any r > 0, then R = 0. If a function of F has a singularity at z0, then R = 0. In the present thesis we will be concerned only with the case in which F = {f(z)} = { n=0infinity afn (z - z0)n} is uniformly bounded in some neighborhood of Z0. As Johnson proves, the radius of regularity R is then given by the formula 1/R=limn→ infinity supf∈F &vbm0;afn&vbm0;1/n where we take R = 0 whenever the righthand expression is infinite.

[发布机构] Rice University