The calculation of integrals encountered in optical diffraction theory

  • R. Barakat
Part of the Topics in Applied Physics book series (TAP, volume 41)


Fast Fourier Transform Point Spread Function Quadrature Formula Gauss Quadrature Quadrature Point 
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Section 2.2

  1. Z.Kopal: Numerical Analysis (Wiley, New York 1961)Google Scholar
  2. P. W. Williams: Numerical Computation (Harper and Row, New York 1972)Google Scholar
  3. P.J. Davis, P. Rabinowitz: Methods of Numerical Integration (Academic Press, New York 1975)Google Scholar
  4. V.I. Krylov: Approximate Evaluation of Integrals (Macmillan, New York 1962)Google Scholar
  5. W. Squire: Integration for Engineers and Scientists (Elsevier, New York 1970)Google Scholar
  6. J. Oliver: The efficiency of extrapolation methods for numerical integration. Numer. Math. 17, 17 (1971)Google Scholar
  7. Z. Kopal: Numerical Analysis (Wiley, New York 1961)Google Scholar
  8. A. H. Stroud: Numerical Quadrature and the Solution of Ordinary Differential Equations (Springer, Berlin, New York, Heidelberg 1973)Google Scholar
  9. A. H. Stroud, D. Secrest: Gaussian Quadrature Formulas (Prentice-Hall, Englewood Cliffs 1966)Google Scholar
  10. L. Filon: On a quadrature formula for trigonometric integrals. Proc. Roy. Soc. Edinburgh 49, 38 (1928)Google Scholar
  11. S. M. Chase, L. D. Fosdick: An algorithm for Filon quadrature. Comm. ACM 12, 453 (1969)Google Scholar
  12. R. Piessens, F. Poleunis: A numerical method for the integration of oscillatory functions. BIT 11, 317 (1972)Google Scholar
  13. E.O. Tuck: A simple Filon-trapezoidal rule. Math. Comp. 21, 239 (1967)Google Scholar
  14. C. J. Tranter: Integral Transforms in Mathematical Physics (Methuen, London 1956) p.76Google Scholar
  15. I. M. Longman: Note on a method for computing infinite integrals of oscillatory functions. Proc. Camb. Phil. Soc. 52, 764 (1956)Google Scholar
  16. P. Cornille: Computation of Hankel transforms. SIAM Rev. 14, 278 (1972)Google Scholar
  17. H. F. Willis: A formula for expanding an integral as a series. Phil. Mag. 39, 455 (1948)Google Scholar
  18. R. Barakat, A. Houston: Reciprocity relations between the transfer function and total illuminance. J. Opt. Soc. Am. 53, 1244 (1963)Google Scholar
  19. R. Barakat, A. Houston: Line spread function and cumulative line spread function for systems with rotational symmetry. J. Opt. Soc. Am. 54, 768 (1964)Google Scholar
  20. B. Tatian: Asymptotic expansions for correcting truncation error in transfer function calculations. J. Opt. Soc. Am. 61, 1214 (1971)Google Scholar
  21. R. Barakat, E. Blackman: The expected value of the edge spread function in the presence of random wavefronts. Opt. Comm. 8, 9 (1973)Google Scholar
  22. V. Mahajan: Asymptotic behavior of diffraction images. Opt. Acta (submitted for publication)Google Scholar

Section 2.3

  1. Most of the material in this section is taken from the author's papers in J. Opt. Soc. Am., Opt. Acta, and J. Appl. Opt., circa 1961–1969.Google Scholar

Section 2.4

  1. E.O. Brigham: The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs 1974)Google Scholar
  2. G. D. Bergland: A guided tour of the fast Fourier transform. IEEE Spectrum 6, 41 (1969)Google Scholar
  3. F. Abramovici: The accurate calculation of Fourier integrals by the Fast Fourier transform technique. J. Compt. Phys. 11, 28 (1973)Google Scholar
  4. A. E. Siegman: Quasi-fast Hankel transform. Opt. Lett. 1, 13 (1977)Google Scholar
  5. L. L. Hope: A fast Gaussian method for Fourier transform evaluation. Proc. IEEE 63, 1353 (1975)Google Scholar

Section 2.5

  1. R. Barakat: Application of the sampling theorem to optical diffraction theory. J. Opt. Soc. Am. 54, 920 (1964)Google Scholar
  2. B.R. Frieden: Image evaluation by use of the sampling theorem. J. Opt. Soc. Am. 56, 1355 (1966)Google Scholar

Copyright information

© Springer-Verlag 1980

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  • R. Barakat

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