Numerical Algorithms

, Volume 72, Issue 1, pp 37–56 | Cite as

On the evaluation of highly oscillatory finite Hankel transform using special functions

  • Zhenhua Xu
  • Shuhuang Xiang
Original Paper


In this paper, an efficient Clenshaw–Curtis–Filon–type method is presented for approximation of the highly oscillatory finite Hankel transform \({{\int }_{0}^{1}}f(x)H_{\nu }^{(1)}(\omega x)dx\), which arises in acoustic and electromagnetic scattering problems. This method is based on Fast Fourier Transform (FFT) and fast computation of the modified moments by using Meijer G–function and Lommel function. Moreover, the method shares the property that the higher the frequency ω, the higher the precision. In particular, for each fixed ω the method is uniformly convergent as N tends to infinity, where (N+1) is the number of Clenshaw–Curtis points c i =(1+ cos(i π/N))/2,i=0,⋯ ,N. Also, the corresponding error bound in inverse powers of ω for this method for the integral is presented. The efficiency and accuracy of the proposed method are illustrated by numerical examples.


Hankel transform Clenshaw–Curtis points Clenshaw–Curtis–Filon–type method Moments Meijer G–function 

Mathematics Subject Classification (2010)

65D32 65D30 


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.College of Mathematics and Information ScienceZhengzhou University of Light IndustryZhengzhouChina
  2. 2.School of Mathematics and StatisticsCentral South UniversityChangshaChina

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