The Legendre function
is considered for large values of
. The asymptotic expansion is in terms of the modified Bessel function
, and holds uniformly with respect to
; the parameter
. The method is based on an integral representation of the Legendre function. A recurrence relation is derived for the coefficients in the expansion, and computable error bounds are derived for the remainders. A comparison is given of the new expansion with an earlier expansion given by Olver, especially with respect to the bounds for the remainders.