Asymptotic expansions for oscillatory integrals using inverse functions.

Journal Article · Fri May 01 04:00:00 EDT 2009 · BIT

DOI:https://doi.org/10.1007/s10543-009-0223-2· OSTI ID:1009331

Lyness, J N; Lottes, J W ^[1]

---

1. Mathematics and Computer Science

We treat finite oscillatory integrals of the form {integral}{sub a}{sup b} F(x)e{sup 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 involved. We simply apply theory involving inverse functions and expansions for a Fourier coefficient {integral}{sub a}{sup b} {phi}(t)e{sup ikt} dt. To this end, we have assembled several results involving inverse functions. Moreover, we have derived a new asymptotic expansion for this integral, valid when {phi}(t) = {Sigma}a{sub j}t{sup {sigma}}j, -1 < {sigma}{sub 1} < {sigma}{sub 2} < {hor_ellipsis}.

• Research Organization: Argonne National Laboratory (ANL)
• Sponsoring Organization: SC
• DOE Contract Number: AC02-06CH11357
• OSTI ID: 1009331
• Report Number(s): ANL/MCS/JA-63328
• Journal Information: BIT, Journal Name: BIT Journal Issue: 2 ; May 2009 Vol. 49; ISSN 0006-3835
• Country of Publication: United States
• Language: ENGLISH

Related Subjects

97 MATHEMATICS AND COMPUTING
ASYMPTOTIC SOLUTIONS
INTEGRALS
OSCILLATIONS