The integral considered here is a loop integral of the form where the integrand has branch points at t =± θ , F ( t 2 , θ 2 ) 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.