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Efficient computation of Cauchy principal value integrals with an irregular oscillator
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DOI：10.1016/j.enganabound.2026.106848.png)
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Abstract
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In this paper, we develop and analyze efficient numerical quadrature methods for the computation of highly oscillatory Cauchy principal value integrals with irregular phases of the form ⨍abf(x)x−τ(x−a)α(b−x)βeiωg(x)dx,τ∈(a,b),ω≫1,where the amplitude function f(x) is analytic in a sufficiently large complex domain containing the interval [a,b]. By analytic continuation, the original integration interval is deformed onto paths of steepest descent, thereby transforming the integral into a sum of several line integrals whose integrands decay exponentially. These resulting integrals are then evaluated efficiently using suitably constructed Gaussian quadrature rules. Finally, error analyses and numerical experiments are presented to illustrate the efficiency and accuracy of the proposed methods.
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