  By application of the theory for second-order linear differential equations
with two turning points developed in [Olver F.W.J., Philos. Trans. Roy. Soc.
London Ser. A 278 (1975), 137-174], uniform asymptotic approximations are
obtained in the first part of this paper for the Lam\'e and Mathieu functions
with a large real parameter. These approximations are expressed in terms of
parabolic cylinder functions, and are uniformly valid in their respective real
open intervals. In all cases explicit bounds are supplied for the error terms
associated with the approximations. Approximations are also obtained for the
large order behaviour for the respective eigenvalues. We restrict ourselves to
a two term uniform approximation. Theoretically more terms in these
approximations could be computed, but the coefficients would be very
complicated. In the second part of this paper we use a simplified method to
obtain uniform asymptotic expansions for these functions. The coefficients are
just polynomials and satisfy simple recurrence relations. The price to pay is
that these asymptotic expansions hold only in a shrinking interval as their
respective parameters become large; this interval however encapsulates all the
interesting oscillatory behaviour of the functions. This simplified method also
gives many terms in asymptotic expansions for these eigenvalues, derived
simultaneously with the coefficients in the function expansions. We provide
rigorous realistic error bounds for the function expansions when truncated and
order estimates for the error when the eigenvalue expansions are truncated.
With this paper we confirm that many of the formal results in the literature
are correct.
