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Analytic Number Theory, Approximation Theory, and Special Functions pp 613–649Cite as

Numerical Integration of Highly Oscillating Functions

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Abstract

Some specific nonstandard methods for numerical integration of highly oscillating functions, mainly based on some contour integration methods and applications of some kinds of Gaussian quadratures, including complex oscillatory weights, are presented in this survey. In particular, Filon-type quadratures for weighted Fourier integrals, exponential-fitting quadrature rules, Gaussian-type quadratures with respect to some complex oscillatory weights, methods for irregular oscillators, as well as two methods for integrals involving highly oscillating Bessel functions are considered. Some numerical examples are included.

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Acknowledgements

The authors were supported in part by the Serbian Ministry of Education, Science and Technological Development (grant number #174015).

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Authors and Affiliations

  1. Mathematical Institute of the Serbian Academy of Sciences and Arts, Belgrade, Serbia

    Gradimir V. Milovanović

  2. Department of Mathematics and Informatics, Faculty of Science, University of Kragujevac, Radoja Domanovića 12, 34000, Kragujevac, Serbia

    Marija P. Stanić

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  1. Gradimir V. Milovanović
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  2. Marija P. Stanić
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Correspondence to Marija P. Stanić .

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Editors and Affiliations

  1. Mathematical Institute, Serbian Academy of Sciences and Arts, Belgrade, Serbia

    Gradimir V. Milovanović

  2. Department of Mathematics, ETH Zürich, Zürich, Switzerland

    Michael Th. Rassias

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Dedicated to Professor Hari M. Srivastava

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Milovanović, G.V., Stanić, M.P. (2014). Numerical Integration of Highly Oscillating Functions. In: Milovanović, G., Rassias, M. (eds) Analytic Number Theory, Approximation Theory, and Special Functions. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0258-3_23

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