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Thread: Holomorphic Extension

  1. #1
    Senior Member roninpro's Avatar
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    Holomorphic Extension

    Hello. I've been trying to show that the function

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

    cannot be holomorphically extended outside the unit disk $\displaystyle D(0,1)$. 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 $\displaystyle z=1, i,\ldots$

    I'd appreciate any suggestions you may have.
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  2. #2
    Junior Member
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    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
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