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On a functional equation for the exponential function of a complex variable

Published online by Cambridge University Press:  18 May 2009

Hiroshi Haruki
Hiroshi Haruki
Affiliation:
University of Waterloo, Ontario, Canada
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The following result is well known in the theory of analytic functions; see [1].

Theorem A. Suppose that f(z) is an entire function of a complex variable z. Then f(z) satisfies the functional equation

where z = x + iy (x, y real), if and only if f(z) = aexp(sz), where a is an arbitrary complex constant and s is an arbitrary real constant.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 12 , Issue 1 , April 1971 , pp. 31 - 34
Copyright
Copyright © Glasgow Mathematical Journal Trust 1971

References

REFERENCES

1.Hille, E., A class of functional equations, Ann. of Math. 29 (1928), 215–222.CrossRefGoogle Scholar
2.Nevanlinna, R. und Polya, G., Unitare Transformationen analytischer Funktionen, Jber. Deutsch. Math. -Verein. 40 (1931), 80 (Aufgabe 103).Google Scholar
3.Pólya, G. und Szego, G., Aufgaben und Lehrsatze aus der Analysis I, p. 94. Berlin-Gottingen-Heidelberg, Springer Verlag, 1954.CrossRefGoogle Scholar
4.Schmidt, H., Losung der Aufgabe 103, Jber. Deutsch. Math. -Verein. 43 (1934), 6–7.Google Scholar