Abstract
By application of the theory for second-order linear differential equations with two turning points developed in the preceding paper, some new asymptotic approximations are obtained for the associated Legendre functions when both the degree n and order m are large. The approximations are expressed in terms of parabolic cylinder functions, and are uniformly valid with respect to
x
∈
(
−
1
,
1
)
and
m
/
(
n
+
1
2
)
∈
[
δ
,
1
+
Δ
]
where δ and ∆ are arbitrary fixed numbers such that 0 < δ < 1 and ∆ > 0. The values of
m
and
n
+ ½ are either both real, or both purely imaginary. In all cases explicit bounds are supplied for the error terms associated with the approximations.
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