Entire function

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January 23rd 2010, 07:32 AM
Arczi1984

Entire function

Hi!
I've one more question this time about entire functions. How I can verify that the below function is entire function? And from where we have the last formula?
http://img46.imageshack.us/img46/5184/th2.th.png
Any help will be highly appreciated.
February 9th 2010, 11:59 AM
Opalg

The element $\exp(\lambda x)$ is by definition the sum of the power series $\textstyle\sum_{n=0}^\infty\lambda^nx^n/n!$ . Thus $\varphi(\lambda) = \textstyle\sum_{n=0}^\infty\lambda^nf(x^n)/n!$ . This is a scalar-valued power series in $\lambda$ with infinite radius of convergence, so it can be differentiated term by term and is analytic throughout $\mathbb{C}$ . If in addition it never takes the value zero, then its logarithm $\psi(\lambda)$ can be defined as an entire function.

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