It is proved that the confluent hypergeometric functionWk,m(z) has no complex zeros when the indexkis real whilemis pure imaginary. Under these conditions, there is an infinite set of positive real zeros with a point of accumulation at zero. The zeros ofWk,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.