Product integrals II: Contour integrals

Computation of contour integrals

Parabolic Marcinkiewicz integrals on product spaces

‎In this paper‎, ‎we study the $L^p$ ($1

The Analysis of Contour Integrals

Recommended by Stephen L. Clark For any n, the contour integral y cosh x ∮ C cosh zs / sinh z − sinhx n 1 dz, s2 −λ, is associated with differential equation d2y x /dx2 λ n n 1 /coshx y x 0. Explicit solutions for n 1 are obtained. For n 1, eigenvalues, eigenfunctions, spectral function, and eigenfunction expansions are explored. This differential equation which does have solution in terms of t...

Contour integrals associated differential equations

C cosh(zs) (sin z − sin x)n+1 dz, y = sinn+1 x ∮ C cosh(zs) (cos z − cos x)n+1 dz, s2 = −λ, and their associated second-order ordinary differential equationswhich are not studied are given. Complete solution is given by calculating residues for n = 1. The eigenvalues and eigenfunctions along with certain boundary conditions are explored for the first contour integral. The contour integral y(x) ...

Numerical contour integration for loop integrals

A fully numerical method to calculate loop integrals, a numerical contour-integration method, is proposed. Loop integrals can be interpreted as a contour integral in a complex plane for an integrand with multi-poles in the plane. Stable and efficient numerical integrations an along appropriate contour can be performed for tensor integrals as well as for scalar ones appearing in loop calculation...