New analytic results for the Zernike circle polynomials from a basic result in the Nijboer-Zernike diffraction theory

A. J. E. M. Janssen

doi:10.2971/jeos.2011.11028

Journal of the European Optical Society - Rapid publications, Vol 6 (2011)

New analytic results for the Zernike circle polynomials from a basic result in the Nijboer-Zernike diffraction theory

A. J. E. M. Janssen

Abstract

Several quantities related to the Zernike circle polynomials admit an expression, via the basic identity in the diffraction theory of Nijboer and Zernike, as an infinite integral involving the product of two or three Bessel functions. In this paper these integrals are identified and evaluated explicitly for the cases of (a)~the expansion coefficients of scaled-and-shifted circle polynomials, (b)~the expansion coefficients of the correlation of two circle polynomials, (c)~the Fourier coefficients occurring in the cosine representation of the circle polynomials.

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