Whittaker functions with both parameters large: uniform approximations in terms of parabolic cylinder functions

Source abstract

Synopsis Asymptotic approximations are derived for the Whittaker functions W κ,μ ( z ), M κ, μ ( z ), W ικ, ιμ ( iz ) and M ικ, ιμ ( iZ ) for large positive values of the parameter μ that are uniform with respect to unrestricted values of the argument z in the open interval (0, ∞), and bounded real values of the ratio κ/μ. The approximations are in terms of parabolic cylinder functions, and in most instances are accompanied by strict error bounds. The results are derived by application of a recently-developed asymptotic theory of second-order differential equations having coalescing turning points, and an extension of the general theory of equations of this kind is also included.