Solution. We can consider the integration from −∞ to ∞ instead. For R > 1, consider the contour that consists of the segment from −R to R and the arc {Reiθ | θ ∈ [0, π]}. Since z2+1 z4+1 decays as |z|−2 on the complex plane as |z| → ∞, this contour integral converges to twice of the original integral when R→∞. The contour encloses two simple poles eπi/4 and e3πi/4 of the function. At eπi/4 the ...

Abstract—In this work, we use the numerical steepest descent path (numerical SDP) method in complex analysis theory to calculate the highly oscillatory physical optics (PO) integral with quadratic phase and amplitude variations on the triangular patch. The Stokes’ phenomenon will occur due to various asymptotic behaviors on different domains. The stationary phase point contributions are careful...

The numerical computation of P{X > x} can be accomplished in a variety of ways. An appealing class of methods may be derived from the contour integral connecting P{X > x} and its Fourier representation. Statisticians have largely focused on deriving saddlepoint approximations for this contour integral. The accuracy of such approximations is generally understood in vague terms only and, perhaps ...

We propose a numerical method using contour integral to solve polynomial eigenvalue problems (PEPs). The method finds eigenvalues contained in a certain domain which is defined by a surrounding integral path. By evaluating the contour integral numerically along the path, the method reduces the original PEP into a small generalized eigenvalue problem, which has the identical eigenvalues in the d...

The notion of a homogeneous contour integral is introduced and used to construct affine integral invariants of convex bodies. Earlier descriptions of this construction rely on a Euclidean structure on the ambient vector space, so the behavior under linear transformations is established after the fact. The description presented here relies only on the linear structure and a Lebesgue measure on t...