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On the functional equations satisfied by modular functions

Published online by Cambridge University Press:  26 February 2010

P. L. Walker
P. L. Walker
Affiliation:
Department of Mathematics, University of Lancaster
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Extract

The Dedekind η-function is defined by

and is well-known to satisfy a functional equation relating η(aτ + b/cτ + d) to η (τ), where a, b, c, d are rational integers with ad – bc = 1—see for instance Iseki [2], and the further references cited there.

Type
Research Article
Information
Mathematika , Volume 25 , Issue 2 , December 1978 , pp. 185 - 190
Copyright
Copyright © University College London 1978

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References

1.Apostol, T. M.. Modular Functions and Dirichlet Series in Number Theory (New York, 1976).CrossRefGoogle Scholar
2.Iseki, S.. “The transformation formula for the Dedekind modular functions and related functional equations.” Duke Math. J., 24 (1957), 653–62.CrossRefGoogle Scholar
3.Serre, J-P.. A Course in Arithmetic (New York, 1973).CrossRefGoogle Scholar
4.Siegel, C.L.. “A simple proof of .” Mathematika, 1 (1954), 4.CrossRefGoogle Scholar
5.Weil, A.. Elliptic functions according to Eisenstein and Kronecker (Berlin, 1976).CrossRefGoogle Scholar
6.Whittaker, E. T. and Watson, G. N.. A course of Modern Analysis. 4th edn. (Cambridge, 1927).Google Scholar