Function-theoretic solution to a class of dual integral equations and an application to diffraction theory
Authors:
Robert A. Schmeltzer and Myrna Lewin
Journal:
Quart. Appl. Math. 21 (1964), 269-283
DOI:
https://doi.org/10.1090/qam/155162
MathSciNet review:
155162
Full-text PDF Free Access
Abstract |
References |
Additional Information
Abstract: Dual integral equations of the type \[ \int _0^\infty {{u^\lambda }f\left ( u \right ){J_\mu }\left ( {ru} \right )du = g\left ( r \right ),0 < r < 1, \\ \int _0^\infty u {{\left ( {{u^2} + {a^2}} \right )}^{ - 1/2}}f\left ( u \right ){J_v}\left ( {ru} \right )du = h\left ( r \right ),} 1 < r < \infty ,\] where $g\left ( r \right )$, $h\left ( r \right )$ are prescribed functions and $f\left ( u \right )$ is to be found, are solved exactly by the application of function-theoretic methods. As an example, a closed-form solution is obtained for the diffraction of an electromagnetic wave by a plane slit.
E. C. Titchmarsh, Introduction to the theory of Fourier integrals, Oxford, 1937, p. 337
- Ida W. Busbridge, Dual Integral Equations, Proc. London Math. Soc. (2) 44 (1938), no. 2, 115–129. MR 1576205, DOI https://doi.org/10.1112/plms/s2-44.2.115
A. S. Peters, Certain dual integral equations and Sonine’s integrals, New York University Report, IMM-285 (1961)
- E. Groschwitz and H. Hönl, Die Beugung elektromagnetischer Wellen am Spalt. I, Z. Physik 131 (1952), 305–319 (German). MR 47531
- C. J. Tranter, A further note on dual integral equations and an application to the diffraction of electromagnetic waves, Quart. J. Mech. Appl. Math. 7 (1954), 317–325. MR 65014, DOI https://doi.org/10.1093/qjmam/7.3.317
R. Müller and K. Westpfahl, A rigorous treatment of the diffraction of electromagnetic waves by a slit,. Zeits. f. Physik 134 (1953) 245–63
- J. C. Gunn, Linearized supersonic aerofoil theory. I, II, Philos. Trans. Roy. Soc. London Ser. A 240 (1947), 327–373. MR 23162, DOI https://doi.org/10.1098/rsta.1947.0005
- A. E. Heins and R. C. MacCamy, A function-theoretic solution of certain integral equations. I, Quart. J. Math. Oxford Ser. (2) 9 (1958), 132–143. MR 102722, DOI https://doi.org/10.1093/qmath/9.1.132
G. N. Watson, Bessel functions, Cambridge, 1934
M. Lewin and R. A. Schmeltzer, A new method for treating supersonic flow past nearly plane wings, to appear in J. Fluid Mechanics
E. C. Titchmarsh, Introduction to the theory of Fourier integrals, Oxford, 1937, p. 337
I. W. Busbridge, Dual integral equations, Proc. London Math. Soc. 44 (1938) 115–29
A. S. Peters, Certain dual integral equations and Sonine’s integrals, New York University Report, IMM-285 (1961)
E. Groschwitz and H. Honl, Diffraction of electromagnetic waves by a slit, Zeit. f. Physik 131 (1952) 305–19
C. J. Tranter, A further note on dual integral equations and an application to the diffraction of electromagnetic waves, Quart. Journ. Mech. Applied Math. 7 (1954) 317–25
R. Müller and K. Westpfahl, A rigorous treatment of the diffraction of electromagnetic waves by a slit,. Zeits. f. Physik 134 (1953) 245–63
J. C. Gunn, Linearized supersonic aerofoil theory, Phil. Trans. A240 (1947) 327
A. E. Heins and R. C. MacCamy, A function-theoretic solution of certain integral equations (I), Quart. J. Math., (2) 9 (1958) 132–43
G. N. Watson, Bessel functions, Cambridge, 1934
M. Lewin and R. A. Schmeltzer, A new method for treating supersonic flow past nearly plane wings, to appear in J. Fluid Mechanics
Additional Information
Article copyright:
© Copyright 1964
American Mathematical Society