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Delange's characterization of the sine function
Published online by Cambridge University Press: 18 May 2009
Extract
In [2], H. Delange gives the following characterization of the sine function.
Theorem A. f(x)=sin x is the only infinitely differentiable real-valued function on the real line such that f'(O)= 1 and
for all real x and n = 0,1,2,….
It is clear that, if f satisfies (1), then the analytic continuation of f is an entire function satisfying
for all z in the complex plane. Hence f is of at most order one and type one. In this note, we prove the following theorem.
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- Copyright © Glasgow Mathematical Journal Trust 1974
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