Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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An analysis of the paraxial wave equation


Authors: Peter A. McCoy and Reza Malek-Madani
Journal: Quart. Appl. Math. 66 (2008), 69-80
MSC (2000): Primary 35L05, 35Q60
DOI: https://doi.org/10.1090/S0033-569X-07-01078-0
Published electronically: December 18, 2007
MathSciNet review: 2396652
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Abstract | References | Similar Articles | Additional Information

Abstract: Function theoretic methods are used to characterize solutions of the paraxial wave equation in an isotropic homogeneous medium in 3-space. A new class of function theoretic solutions whose singularities are manifested as sectionally analytic functions is constructed via integral transforms.


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  • 1. Larry C. Andrews, Special functions for engineers and applied mathematicians, Macmillan Co., New York, 1985. MR 779819
  • 2. H. Bateman, Higher Transcendental Functions, vol. 2, Compiled by the Staff of the Bateman Manuscript Proj., McGraw-Hill, New York, 1953.
  • 3. Stefan Bergman, Integral operators in the theory of linear partial differential equations, Second revised printing. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 23, Springer-Verlag New York Inc., New York, 1969. MR 0239239
  • 4. H. Begher & R.P. Gilbert, Transmutations, Transformations and Kernel Functions, Pitman Monographs & Surveys in Pure and Appl. Math., vols. 58-59, New York, 1992.
  • 5. W.B. Colson et. al., Laser Handbook: Free Electron Lasers, Elsevier Science LTD, San Diego & New York, 1991.
  • 6. D.L. Colton & R. Kress, Inverse Acoustic and Electromagnetic Scattering, Appl. Math Sci., vol. 93, Springer-Verlag, New York, 1992.
  • 7. Robert P. Gilbert, Function theoretic methods in partial differential equations, Mathematics in Science and Engineering, Vol. 54, Academic Press, New York-London, 1969. MR 0241789
  • 8. Robert P. Gilbert and Roger G. Newton (eds.), Analytic methods in mathematical physics, Gordon and Breach Science Publishers, New York-London-Paris, 1970. Based on the Symposium held at Indiana University, Bloomington, Indiana, June 2–6, 1968. MR 0327404
  • 9. Jianming Jin, The finite element method in electromagnetics, 2nd ed., Wiley-Interscience [John Wiley & Sons], New York, 2002. MR 1903357
  • 10. R. Victor Jones, ``On Classical Electromagnetic Fields (cont.)'', http://people.seas.harvard.edu/~jones/ap216/lectures/ls_1/ls1_u3/ls1_unit_3.html
  • 11. Manfred Kracht and Erwin Kreyszig, Methods of complex analysis in partial differential equations with applications, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1988. A Wiley-Interscience Publication. MR 941372
  • 12. M.Z. v. Krzywoblocki, Bergman's and Gilbert's operators in elasticity, electromagnetism, fluid dynamics, wave mechanics, Analytic Methods in Mathematical Physics, Gordon & Breach Scientific Publishers, New York (1970) 207-247.
  • 13. Peter A. McCoy, On radiating solutions to the Helmholtz equation and inverse scattering, Appl. Anal. 77 (2001), no. 3-4, 319–326. MR 1975738, https://doi.org/10.1080/00036810108840911
  • 14. Peter A. McCoy, Electromagnetic field singularities, J. Math. Anal. Appl. 275 (2002), no. 2, 761–770. MR 1943778, https://doi.org/10.1016/S0022-247X(02)00420-1
  • 15. Claus Müller, Foundations of the mathematical theory of electromagnetic waves, Revised and enlarged translation from the German. Die Grundlehren der mathematischen Wissenschaften, Band 155, Springer-Verlag, New York-Heidelberg, 1969. MR 0253638
  • 16. A.E. Siegman, Lasers, University Science Books, Mill Valley, CA, 1986.
  • 17. P. Sprangle et. al., Atmospheric Propagation of Ultrashort Laser Pulses, 6th Directed Energy Symposium, Albuquerque, http://www.deps.org/DEPSpages/DEsymp03.html
  • 18. P. Sprangle et. al., Focusing of Intense Pulses Using Plasma Channels, AIP Conf. Proc., vol. 647(1), pp. 664-673, Dec. 2002.
  • 19. G. Szego, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, Providence, 1967.
  • 20. E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. MR 1424469

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Additional Information

Peter A. McCoy
Affiliation: Department of Mathematics, U.S. Naval Academy, Annapolis, Maryland 21402-5002

Reza Malek-Madani
Affiliation: Department of Mathematics, U.S. Naval Academy, Annapolis, Maryland 21402-5002

DOI: https://doi.org/10.1090/S0033-569X-07-01078-0
Received by editor(s): May 31, 2006
Published electronically: December 18, 2007
Article copyright: © Copyright 2007 Brown University


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