By application of the theory for second order linear differential equations with a turning point and a regular (double pole) singularity developed by W. G. C. Boyd
and the author [ibid. 17, 422-450 (1986; Zbl 0591.34048
)] uniform asymptotic expansions are obtained for prolate spheroidal functions for large
. The results are uniformly valid for
, where A, A’ and A” are arbitrary real constants such that
. An asymptotic relationship between
and the characteristic component
is then derived from the approximations for the spheroidal functions. All the error terms associated with the approximations have explicit bounds given.