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Math Help - Entire function

  1. #1
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    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?

    Any help will be highly appreciated.
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  2. #2
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    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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