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The expansion of an arbitrary function in a series of conal or toroidal functions

Published online by Cambridge University Press:  24 October 2008

D. P. Banerji
Affiliation:
A.M. College, Mymensingh, Bengal

Extract

1. Particular series of conal and toroidal functions have been studied by various authors*, but so far no method has been given of expanding an arbitrary function in such a series. Ordinary methods fail, since conal functions (and also toroidal functions) are not orthogonal, although they form a complete set. In this paper I give an expansion theorem.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1938

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References

* Cf. Ganesh, Prasad, Bull. Calcutta Math. Soc. 23 (1931), 115;Google ScholarShabde, N. G., Bull. Calcutta Math. Soc. 23 (1931), 155.Google Scholar

Hobson, , The theory of spherical and ellipsoidal harmonics (Cambridge, 1931), p. 451,Google Scholar equation (c).

Whittaker, and Watson, , Modern analysis, 4th ed. (Cambridge, 1927), p. 229.Google Scholar

* Hobson, op. cit., p. 267, equation (130).

Watson, , Theory of Bessel functions (Cambridge, 1922),Google Scholar chapter 19; cf. also the methods employed by Levinson, , Annals of Math. 37 (1936), 919936.CrossRefGoogle Scholar

A is said to be closed if f nf and Af ng imply that Af is defined and equal to g, the limits being taken in the metric.

* See Rellich, , Math. Ann. 110 (1934), 342–56.CrossRefGoogle Scholar