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On Mellin's inversion formula

Published online by Cambridge University Press:  24 October 2008

J. C. Burkill
Affiliation:
Trinity College

Extract

The extension of Mellin's inversion formula expressed by the equations has been considered by Fowler who shows that some form of Stieltjes integral is essential to Poincaré's proof of the necessity of the quantum hypothesis. Fowler confines his discussion to a restricted type of function φ (y) which is sufficient for the physical problem. It will be proved here that the formulae hold with a general Stieltjes integral in the first equation.

Type
Articles
Copyright
Copyright © Cambridge Philosophical Society 1926

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References

* Proc. Royal Soc. (A), vol. 99 (1921), pp. 462471CrossRefGoogle Scholar. An account of Mellin's formula in the ordinary form (without Stieltjes integrals) is given by Hardy, , Messenger of Math. vol. 47 (1918), pp. 178184 and vol. 50 (1921), pp. 166–171.Google Scholar

The expression in Stieltjes integrals of the inversion formulae of Fourier and Hankel,” Proc. London Math Soc. (unpublished).Google Scholar

* Oppenheim, A., “Some identities in the theory of numbers,” Proc. London Math. Soc. (Records), vol. 24 (1925), p. xxiii. I am indebted to Mr Oppenheim for sending me in MS. the part of his work dealing with discontinuous factors.Google Scholar

Watson, , Bessel Functions, p. 198.Google Scholar