# Holomorphic Extension

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• April 4th 2011, 10:27 PM
roninpro
Holomorphic Extension
Hello. I've been trying to show that the function

$\displaystyle f(z)=\sum_{n=1}^\infty \frac{z^{n!}}{n}$

cannot be holomorphically extended outside the unit disk $D(0,1)$. I'd like to show that the function "blows up" at the boundary, but it isn't clear to me how to do that, except for simple points like $z=1, i,\ldots$

I'd appreciate any suggestions you may have.
• April 5th 2011, 05:00 PM
choovuck
take z to be a point on the unit circle with rational argument, i.e. z=exp(ik/m). then eventually all those terms become 1/n...... u can do the rest :)