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Bessel-zernike discrete variable representation basis.

Charles Cerjan1

  • 1Lawrence Livermore National Laboratory, Livermore California 94550, USA.

The Journal of Physical Chemistry. A
|April 21, 2006
PubMed
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This study demonstrates how Zernike polynomials, a type of Jacobi polynomial, can be used for Bessel function expansions. This method simplifies generating series identities and efficiently evaluating Hankel transforms.

Area of Science:

  • Mathematical Physics
  • Special Functions

Background:

  • Bessel functions are crucial in various scientific fields.
  • Orthogonal polynomials offer powerful tools for function approximation and analysis.
  • Zernike polynomials are a specific class of orthogonal polynomials with unique properties.

Purpose of the Study:

  • To establish the connection between Bessel discrete variable basis expansions and Zernike polynomials.
  • To demonstrate the utility of Zernike polynomials for function series expansions.
  • To explore the application of Zernike polynomials in evaluating Hankel transforms.

Main Methods:

  • Demonstrating the relationship between Bessel basis expansions and Jacobi polynomials.
  • Utilizing Zernike polynomials for series expansions of functions over the unit interval.

Related Experiment Videos

  • Applying Zernike expansions to Bessel functions to derive series identities.

    • Main Results:

    • A direct connection is shown between Bessel discrete variable basis expansion and Zernike polynomials.
    • Zernike polynomials provide an alternative and effective method for series expansions.
    • Expressing Bessel functions using Zernike expansions simplifies the generation of series identities.
    • Zernike polynomials enable efficient computation of Hankel transforms for specific function types.

    Conclusions:

    • Zernike polynomials offer a valuable framework for analyzing Bessel functions.
    • The use of Zernike polynomials streamlines the derivation of mathematical identities.
    • This approach provides an efficient computational method for Hankel transforms.