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Complex Analysis-Series

  1. Aug 10, 2010 #1
    The problem statement, all variables and given/known data
    Find the analytic regions of the next functions:

    A. [tex] f(z)=\sum_{n=1}^{\infty} [ \frac{z(z+n)}{n}]^n [/tex]
    B. [tex] f(z)=\sum_{n=1}^{\infty} 2^{-n^2 z} \cdot n^n [/tex]

    2. Relevant equations
    3. The attempt at a solution

    In the first one: I've tried writing : [tex] f(z)= \sum \frac{z^{2n}}{n^n} + \sum z^n [/tex]
    and the second element in the sum converges iff |z|<1... Is it enough?

    About B: We can write this series as: [tex] \sum [ \frac{n}{2^{nz}}]^n [/tex] ... But I don't think it helps us...

    Hope you'll be able to help me


    TNX!
     
  2. jcsd
  3. Aug 10, 2010 #2
    The first series ultimately grows like z^n exp(z) & hence converges for |z|<1.
    The second problem can be tackled similarly.
     
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