Abstract
We describe the two uniform asymptotic expansions which already exist and which are commonly used in asymptotic electromagnetic theories, when a pole approaches the saddle point. Then, we establish a simple relation between these two expansions. By analyzing this relation, we give the general form of any asymptotic expansion with a pole near the saddle point. By using diffraction integrals, we give explicit examples and compare the corresponding numerical computations. Then we conclude by examining the specific problem of many coalescing poles singularities.
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