Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Integrals with a large parameter: Legendre functions of large degree and fixed order

F. Ursella1

a1 Department of Mathematics, University of Manchester, Manchester M13 9PL


Suppose that a function f(z, n) depends on a large parameter n. A proposed expression g(z, n) is an asymptotic approximation for f(z, n) if it can be shown that the error (i.e. the difference between f(z, n) and g(z, n)) is small of an appropriate order when n → ∞. Effective error bounds are particularly useful in numerical work with asymptotic expansions. Most of the existing derivations of error bounds involve complicated calculations.

(Received September 07 1983)

(Revised October 17 1983)