The Journal of the Indian Mathematical Society

V. Ganapathi Iyer

Abstract

This is an attempt at an exhaustive classification of infinite series whose terms are complex numbers, according to the manner of distribution of the limiting values of the series when the terms of the series are re-arranged so as to form another series. By the limiting values of the series are meant the limit points of the sequence obtained by taking the sum to first n terms of the series in any particular rearrangement. A definite arrangement of the terms of the series being presupposed, the series converges or diverges according as the sequence has a unique finite or infinite limit; otherwise the series oscillates.

Full Text

     

V. Ganapathi Iyer

Abstract

In what follows it is supposed that the half-plane of convergence is σ>0 and that s=ξ→0 through positive real values. The object of this paper is to prove for the series (1) the analogue of the following theorems for Dirichlets' series.

Full Text

     

V. Ganapathi Iyer ¹

Affiliations
1 Madras University, IN

Abstract

Let {f_n(z)}, n=1, 2, . . . , be a family of functions holomorphic in the interior of a domain D. The family is said to be normal in D when every sub-sequence selected from the family contains another converging uniformly in any closed region lying in D.

Full Text