Asymptotics of Prolate Spheroidal Wave Functions

Uniform asymptotic approximations are obtained for the prolate spheroidal wave functions, in the high-frequency case. The results are obtained by an application of certain existing asymptotic solutions of differential equations, and involve elementary, Bessel, and parabolic cylinder functions. An asymptotic relationship between the prolate spheroidal equation separation parameter and the other parameters is also obtained, and error bounds are available for all approximations