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Integrals with nearly coincident branch points: Gegenbauer polynomials of large degree

The integral considered here is a loop integral of the formwhere the integrand has branch points at t=±θ, F(t2, θ2) is an analytic function of its arguments and N is a large positive parameter. When θ is not small, its complete asymptotic expansion can be found by standard techniques. When θ is small, the branch points are nearly coincident, and it will be shown that there is a uniform asymptotic expansion involving Bessel functions of argument Nθ. An inequality will be established and will be used to show that the expansion is valid in a region including a disc |θ|≤m of the complex θ-plane, where m does not tend to 0, when N tends to ∞. The proof of this inequality uses the maximum-modulus principle of complex function theory.