Asymptotic expansions of Mellin convolution integrals: An oscillatory case

Asymptotic expansions of oscillatory integrals with complex phase

We consider saddle point integrals in d variables whose phase functions are neither real nor purely imaginary. Results analogous to those for Laplace (real phase) and Fourier (imaginary phase) integrals hold whenever the phase function is analytic and nondegenerate. These results generalize what is well known for integrals of Laplace and Fourier type. The proofs are via contour shifting in comp...

Asymptotic Expansions for Oscillatory Integrals Using Inverse Functions

We treat finite oscillatory integrals of the form ∫ b a F (x) exp[ikG(x)]dx in which both F and G are real on the real line, are analytic over the open integration interval, and may have algebraic singularities at either or both interval end points. For many of these, we establish asymptotic expansions in inverse powers of k. No appeal to the theories of stationary phase or steepest descent is ...

Asymptotic Expansions for Integrals

Asymptotic Expansions of Symmetric Standard Elliptic Integrals

Symmetric standard elliptic integrals are considered when one of their parameters is larger than the others. Distributional approach is used for deriving five convergent expansions of these integrals in inverse powers of the respective five possible asymptotic parameters. Four of these expansions involve also a logarithmic term in the asymptotic variable. Coefficients of these expansions are ob...

Resolution of Singularities, Asymptotic Expansions of Oscillatory Integrals, and Related Phenomena

Here φ(x) is a smooth bump function defined on a neighborhood of the origin and λ is a real parameter whose absolute value we assume to be large. If ∇f(0) 6= 0, then by repeated integrations by parts (see [S] Ch 8 for example), for any N one has an estimate |Iλ| < Cf,φ,N |λ| −N for appropriate constants Cf,φ,N . In the case where ∇f(0) = 0, that is, when f has a critical point at the origin, it...