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
Full-text PDF Free Access
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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.
References
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References
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- H. Bateman, Higher Transcendental Functions, vol. 2, Compiled by the Staff of the Bateman Manuscript Proj., McGraw-Hill, New York, 1953.
- S. Bergman, Integral Operators in the Theory of Linear Partial Differential Equations, 2nd rev. printing, Springer-Verlag, New York, 1969. MR 0239239 (39:596)
- H. Begher & R.P. Gilbert, Transmutations, Transformations and Kernel Functions, Pitman Monographs & Surveys in Pure and Appl. Math., vols. 58-59, New York, 1992.
- W.B. Colson et. al., Laser Handbook: Free Electron Lasers, Elsevier Science LTD, San Diego & New York, 1991.
- D.L. Colton & R. Kress, Inverse Acoustic and Electromagnetic Scattering, Appl. Math Sci., vol. 93, Springer-Verlag, New York, 1992.
- R.P. Gilbert, Function Theoretic Methods in Partial Differential Equations, Math. in Sci. & Engr., 54, Academic Press, New York, 1969. MR 0241789 (39:3127)
- R.P. Gilbert & R.G. Newton (eds.), Analytic Methods in Mathematical Physics, Gordon & Breach Sci. Pub., New York, 1970. MR 0327404 (48:5746)
- J. Jin, The Finite Element Method in Electromagnetics, John Wiley & Sons, New York, 1993. MR 1903357 (2004b:78019)
- R. Victor Jones, “On Classical Electromagnetic Fields (cont.)”, http://people.seas.harvard.edu/~jones/ap216/lectures/ls_{1}/ls1_{u}3/ls1_{u}nit_{3}.html
- M. Kracht & E. O. Kreyszig, Methods of Complex Analysis in Partial Differential Equations with Applications, Can. Math. Soc. Series of Monographs & Advanced Texts, Wiley-Interscience Publication, New York, 1988. MR 941372 (89f:35002)
- 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.
- P.A. McCoy, On Radiating Solutions to the Helmholtz Equation and Inverse Scattering, Appl. Analysis, vol. 77, nos. 3-4 (2001) 319-326. MR 1975738 (2004a:35033)
- P.A. McCoy, Electromagnetic Field Singularities, J. Math. Anal. & Appl, vol. 275 (2002), pp. 761-770. MR 1943778 (2003m:78014)
- C. Muller, Mathematical Theory of Electromagnetic Waves, Die Grundlehren der mathematischen Wissenschaften, rev. ed., vol. 155, Springer-Verlag, New York, 1969. MR 0253638 (40:6852)
- A.E. Siegman, Lasers, University Science Books, Mill Valley, CA, 1986.
- P. Sprangle et. al., Atmospheric Propagation of Ultrashort Laser Pulses, 6th Directed Energy Symposium, Albuquerque, http://www.deps.org/DEPSpages/DEsymp03.html
- P. Sprangle et. al., Focusing of Intense Pulses Using Plasma Channels, AIP Conf. Proc., vol. 647(1), pp. 664-673, Dec. 2002.
- G. Szego, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, Providence, 1967.
- E.T. Whittaker & G.N. Watson, A Course of Modern Analysis, Cambridge University Press, 4th ed. (reprinted) New York, 1969. MR 1424469 (97k:01072)
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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
MR Author ID:
118725
Received by editor(s):
May 31, 2006
Published electronically:
December 18, 2007
Article copyright:
© Copyright 2007
Brown University
