Addition theorems for spherical waves

Authors:
Bernard Friedman and Joy Russek

Journal:
Quart. Appl. Math. **12** (1954), 13-23

MSC:
Primary 33.0X

DOI:
https://doi.org/10.1090/qam/60649

MathSciNet review:
60649

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Abstract: Expansions or ``addition theorems'' for the spherical wave functions , , and , with reference to the origin , have been obtained in terms of spherical wave functions with reference to the origin , where , has the coordinates with respect to .

**[1]**See for example, G. N. Watson,*Bessel functions*, Cambridge, 1948, pp. 359-361, or Stratton,*Electromagnetic theory*, McGraw Hill, 1941, pp. 372-374.**[2]**V. Twersky, J. Acoust. Soc. Am.**24**, 42 (1952); J. Appl. Phys.**23**, 407, 1099 (1952). MR**0043718****[3]**J. A. Stratton;*Electromagnetic theory*, McGraw Hill, N. Y., 1941, p. 409.**[4]**Infeld and Hull, Reviews Modern Physics,**23**, 54 (1951).**[5]**J. A. Stratton,*Electromagnetic theory*, McGraw Hill, 1941, p. 410.**[6]***Ibid.*, p. 578.**[7]***Ibid.*, p. 414.**[8]**A similar integral representation for has been obtained by Satô, Yasuo, Bull. Earthquake Res. Institute, Tokyo,**28**, 1-22 and 175-217 (1950). In this paper Yasuo obtains an addition theorem for also; however, his expansion is valid only for a translation of the origin along the -axis. The addition theorem he obtains contains powers of instead of the Bessel functions , and the coefficients of his expansion are not given by a general formula but must be evaluated by a recurrence relation.**[9]**J. A. Stratton,*Electromagnetic theory*, McGraw Hill, 1941, p. 414.**[10]**Magnus and Oberhettinger,*Special functions of mathematical physics*, Chelsea, 1949, p. 64-66.

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DOI:
https://doi.org/10.1090/qam/60649

Article copyright:
© Copyright 1954
American Mathematical Society