Suppose I have an expression of the form

$$f(z) := f_1(z)+f_2(z)$$ ($f,f_1,f_2$ can e.g. be integrals) with $f_1$ convergent in the region $R_1=\{\Re(z)>-1\}$ and $f_2$ convergent in the region $R_2=\{\Re(z)<1\}$.

Let $R= R_1\cap R_2$. Then, to my knowledge, $f$ is convergent in $R$. My question is when is it possible to meromorphically extend $f$ to a larger region, say all of $\mathbb{C}$?

  • 1
    $\begingroup$ You have a continuation across $\Re z = -1$ if and only if $f_1$ has one, and a continuation across $\Re z = 1$ if and only if $f_2$ has one. $\endgroup$ Nov 21, 2013 at 15:21
  • $\begingroup$ Makes sense, I suspected this. Thanks for the answer. $\endgroup$ Nov 21, 2013 at 16:55


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