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On the circle polynomials of Zernike and related orthogonal sets

Published online by Cambridge University Press:  24 October 2008

A. B. Bhatia  and
E. Wolf
A. B. Bhatia
Affiliation:
Department of Mathematical PhysicsUniversity of Edinburgh
E. Wolf
Affiliation:
Department of Mathematical PhysicsUniversity of Edinburgh
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Abstract

The paper is concerned with the construction of polynomials in two variables, which form a complete orthogonal set for the interior of the unit circle and which are ‘invariant in form’ with respect to rotations of axes about the origin of coordinates. It is found that though there exist an infinity of such sets there is only one set which in addition has certain simple properties strictly analogous to that of Legendre polynomials. This set is found to be identical with the set of the circle polynomials of Zernike which play an important part in the theory of phase contrast and in the Nijboer-Zernike diffraction theory of optical aberrations.

The results make it possible to derive explicit expressions for the Zernike polynomials in a simple, systematic manner. The method employed may also be used to derive other orthogonal sets. One new set is investigated, and the generating functions for this set and for the Zernike polynomials are also given.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 50 , Issue 1 , January 1954 , pp. 40 - 48
Copyright
Copyright © Cambridge Philosophical Society 1954

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References

REFERENCES

(1)Born, M.Natural philosophy of cause and chance. (Oxford, 1949).Google Scholar
(2)Courant, R. and Hilbert, D.Methoden der Mathematischen Physik, vol. 1, 2nd ed. (Berlin, 1931).Google Scholar
(3)Kemble, E.The fundamental principles of quantum mechanics, 1st ed. (New York, 1937).Google Scholar
(4)Madelung, E.Die Mathematischen Hilfsmittel des Physikers (Berlin, 1936).Google Scholar
(5)Nijboer, B. R. A. Thesis, University of Groningen (1942).Google Scholar
(6)Nijboer, B. R. A.Physica, 's Grav., 13 (1947), 605.Google Scholar
(7)Zernike, F.Physica, 'a Grav., 1 (1934), 689.CrossRefGoogle Scholar
(8)Zernike, F. and Brinkman, H. C.Proc. Akad. Sci. Amst. 38, no. 2 (1935).Google Scholar
(9)Zernike, F. and Nijboer, B. R. A.Contribution in La Théorie des Images Optiques (Paris, 1949. Editions de la Revue d'Optique), p. 227.Google Scholar