I was reviewing for my Complex Analysis exam and stumbled upon this question. I'm having trouble proving that the coefficients of a power series are equal to 0 if the radius of convergence is larger than 0 and f(z)=0. Intuitively to me it seems clear than an should equal to 0 if f(z)=0, but I don't know how to formally prove or show this. Any help, tips, pointers, and suggestions would be appreciated. Note that this an elementary undergraduate complex analysis course, so nothing too advanced please. I have attached the question as a picture. Thank you.


  • $\begingroup$ $a_n = \frac{f^{(n)}(z_0)}{n!}$ $\endgroup$ – Friedrich Philipp Apr 2 '17 at 23:04

If $f(z)=0$ for any $z$ within your radius, then for any $z$ in the disc you have $$ f'(z)=0\\ f''(z)=0\\ \vdots $$ Does that clarify?

  • $\begingroup$ Yes that makes sense. $\endgroup$ – kemb Apr 2 '17 at 23:09

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