= Gaussian saddle-pole transition
{c}
For a simple <saddle point>, a nearby <pole> coalesces with its contributing region when their separation is of order the inverse square root of the large parameter. A local quadratic phase coordinate changes the singular part to a <Gaussian pole integral>. Adding any residue crossed by the <contour deformation> gives a <Faddeeva function> rather than two separately singular approximations. The regular part of the transformed amplitude remains smaller by the ordinary saddle width.
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