Stability of an idealized atmosphere. II. Zeros of the confluent hypergeometric function

Abstract

It is proved that the confluent, hypergeometric function Wk,m(z) has no complex zeros when the index k is real while m is pure imaginary. Under these conditions, there is an infinite set of positive real zeros with a point of accumulation at zero. The zeros of Wk,m(z) are related to the stability of a model incompressible atmosphere, with density decreasing exponentially and horizontal wind velocity increasing linearly with height. The nonexistence of complex zeros indicates that this model atmosphere should be stable. The stability is rigorously proved in an accompanying paper by Case.