JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2017, Vol. 52 ›› Issue (10): 50-55.doi: 10.6040/j.issn.1671-9352.0.2016.558

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The steepest descent method for Fourier integrals involving algebraic and logarithmic singular factors

KONG Yi-ting, WANG Tong-ke*   

  1. School of Mathematical Sciences, Tianjin Normal University, Tianjin 300387, China
  • Received:2016-11-25 Online:2017-10-20 Published:2017-10-12

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Abstract: For the oscillatory Fourier integral over finite or semi-infinite interval that has algebraic and logarithmic singularities at the endpoints, this paper converts the oscillatory factor to fast decaying factor by deforming the path of integration into complex plane such that the integral never oscillates along the new path. For the transformed singular integrals over semi-infinite interval, the modified Gauss-Legendre quadrature formula can be used to evaluate them efficiently. Numerical examples verify the correctness of the theoretical analysis and the high accuracy of the method.

Key words: oscillatory Fourier integral, the steepest descent method, finite or semi-infinite interval, algebraic and logarithmic singularity

CLC Number: 

  • O174.22
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