圆环上半纯的典型实照函数

<正> §1.引言1932年 Rogosinski 首先研究了单位圆 E:|z|<1内正则的典型实照函数,这种函数的全体成一函数族 T_r(E)假如 f(z)∈T_r(E),那末 f(z)=z+a_2z~2+…在|z|<1是正则的,且满足条件

FUNCTIONS TYPICALLY REAL AND MEROMORPHIC IN A CIRCULAR RING

In the circular ring R_q:q<|z|<1 let the function f(z) be meromorphic andsatisfy the condition(?)The class of all such functions f(z)will be denoted by T_r(R_q).Let the poles p_i of f(z)∈Tr(R_q)be arranged so that(?)Let the Laurent expansion for f(z)in the ring p′<|z|1)is odd,where(?)when n is even;and(?)when n(>1)is odd;(?) when n is even;(?)when n(>1)is odd.All the above estimates are precise.Theorem 6.Corresponding to a function f(z)of T_r(R_q)there exists a functiong(z)regular and typically real in R_q,such that(?)where the p_j(j=O,±1,±2,...)are the poles of f(z)in R_q.From this result we can deduce an integral representation of f(z)∈T_r(R_q).