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Abhandlungen
Clebsch,Über die Reflexion an einer Kugelfläche, J. f. Math. 61, p. 195 (1863).
If the angle is β/m, m any positive integer, the ordinary method of images gives the solution. If the angle isnπ/m, m andn any positive integers, the method of images in a Riemann's Space is applied. [Carslaw,Some Multiform Solutions of the Partial Differential Equations of Physical Mathematics-and their Applications, Proc. Lond. Math. Soc. (1) 30, p. 135 (1899)]. In a later communication [Proc. Lond. Math. Soc. (2) 8, p. 365 (1912)] I have pointed out that a suitable solution of the equation of period 2α leads at once to the solution of the problem for the case of two planes intersecting at any angle α.
Compare also a paper by Sommerfeld in Jahresb. D. Math. Ver. 22 (1913), entitledDie Greensche Funktion der Schwingungsgleichung.
Cf. Heine,Handbuch der Kugelfunktionen, Bd. 1, p. 346; Macdonald, Proc. Lond. Math. Soc. (1) 32, p. 157 (1900). In this paper is taken as the Second Solution of Bessel's Equation of then th order.
Cf. Dougall,The Determination of Green's Function by means of Cylindrical or Spherical Harmonics, Proc. Edin. Math. Soc.18, p. 33 (1900).
Nielsen,Handbuch der Cylinderfunktionen, p. 7.
Cf. Nielsen,loc. cit. Handbuch der Cylinderfunktionen, p. 11.
Cf. Heine, loc. cit.Handbuch der Kugelfunktionen, Bd. 1, p. 346; Vol. 1, p. 175. Hobson, Phil. Trans.187 (A), p. 489 (1896).
The Zonal HarmonicP n(μ) can be defined by the Hypergeometric SeriesF(−n, n+1, 1, 1−μ/2). With this definition the relationP n(μ)=(−1)n P n(−μ) follows for an integral value ofn. Hobson,loc. cit. Phil. Trans.187 (A), p. 463 (1896).
Whenx is large, approximately. Cf. Nielsen,loc. cit. Phil. Trans.187 (A), p. 489 (1896). p. 154.
This is a special case of a Theorem proved by Macdonald,Zeroes of the Spherical Harmonic P mn (μ) considered as a Function of n, Proc. Lond. Math. Soc. (1) 31, p. 265 (1900). That the zeroes are all real, follows from the integral That they are all distinct, from the integraln satisfying the same condition as above, since this shows that Cannot both vanish at the same time.
Cf. Hobson,, p. 473.
Cf. Macdonald,, Proc. Lond. Math. Soc. (1) 31, p. 276 (1900).
Cf. Hobson,, p. 489. His results hold forn, real or imaginary.
It is known that the zeroes ofP −mn (μ) are all real and distinct (Cf. Macdonald, Proc. Lond. Math. Soc. (1) 31, p. 265 (1900)). It would be interesting to have a proof of the corresponding result for the functiond/dμP −mn (μ). With the other boundary condition usually associated with these Green's Functions, viz.,u=0, we are brought to a result similar to (18), the denominator involving 143-1, and we obtain the Infinite Series form immediately.
Macdonald, Proc. Lond. Math. Soc. (1) 31, p. 274 (1899).
Cf. Hobson, Phil. Trans. (A) 187, p. 473 (33), (1896).
Cf. Trans. Camb. Phil. Soc. 18, p. 292 (1900).
Phil. Mag. (6) 20, p. 690 (1910).
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Carslaw, H.S. The scattering of sound waves by a cone. Math. Ann. 75, 133–147 (1914). https://doi.org/10.1007/BF01564524
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DOI: https://doi.org/10.1007/BF01564524