Vol. 50, No. 2, 1974

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Limit sets of power series outside the circles of convergence

Charles Kam-Tai Chui and Milton N. Parnes

Vol. 50 (1974), No. 2, 403–423

Let U denote the open unit disc in the complex plane C. A power series of a complex variable with center at the origin and radius of convergence equal to one will be called a powe7 series in U. It is well-known that the behaviors of a power series outside its circle of convergence are quite irregular. In particular, it is proved that for each complex number z,|z| > 1, and for every closed set E in C, there is a power series in U whose limit set at z is E. The power series Pα(z) = (1 einα)znreal, are studied in this paper. Their peculiar properties seem to suggest that it might be fruitful to study the irregularities of power series outside their circles of convergence by probability theory. To study Pα(z), results in diophantine approximation are obtained.

Mathematical Subject Classification
Primary: 30A12
Received: 16 February 1973
Published: 1 February 1974
Charles Kam-Tai Chui
Milton N. Parnes