Hostname: page-component-848d4c4894-hfldf Total loading time: 0 Render date: 2024-04-30T14:29:48.371Z Has data issue: false hasContentIssue false
  • English
  • Français

Theorems connecting different classes of Self-Reciprocal Functions

Published online by Cambridge University Press:  20 January 2009

Brij Mohan Mehrotra
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

1. The question, “How, from a given function which is self-reciprocal for a transform of a particular order, can we construct other functions which are self-reciprocal for transforms of different orders?” was first raised by Hardy and Titchmarsh who gave some rules for constructing such functions. Following their method, I have shown, in a recent paper, that there are certain general theorems of the following type:—

If f (x) is its own Jμtransform, g (x) is its own Jv transform. In this note I add a few more such theorems, the interest lying mainly in the results themselves and not in a rigorous proof thereof; and hence only the formal procedure is given here.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 4 , Issue 1 , March 1934 , pp. 53 - 56
Copyright
Copyright © Edinburgh Mathematical Society 1934

References

REFERENCES

1.Bafceman, H.. “Report on the history and present state of the theory of Integral Equations.” Reports of the British Assoc. (Sheffield, 1910), 61.Google Scholar
2.Hardy, G. H.. “Further notes on Mellin's Inversion Formula.” Messenger of Math., 50 (1921), 165–171.Google Scholar
3.Hardy, G. H. and Titchmarsh, E. C.. “Self-Reciprocal Functions.” Quart. Journal of Math., Oxford Series 1 (1930), 196–231.CrossRefGoogle Scholar
4.Hardy, G. H. and Titchmarsh, E. C.. “Formulae connecting different classes of SelfReciprocal Functions.” Proa. London Math. Soc, II (33) (1931), 225'232.Google Scholar
5.Mehrotra, B. M.. “Some Theorems on Self-Reciprocal Functions.” Proc. London Math. Soc, II (34) (1932), 231'240.Google Scholar
6.Pincherle, S.. “Studi sopra alcune operazioni funzionali.” Mem. di Bologna (4), 7 (1886), 393'442.Google Scholar
7.Watson, G. N.. Theory of Bessl Functions. Cambridge 1922.Google Scholar