I'm wondering about uniform convergence. We're looking at it in my complex analysis class. We are using uniform convergence of a series of functions, to say that we can interchange integration of the sum, that is: [itex]\int\sum b_{j}z^{j}dz[/itex]=[itex]\sum\int b_{j}z^{j}dz[/itex]=[itex]\int f(z)dz[/itex](adsbygoogle = window.adsbygoogle || []).push({});

On an intuitive level I don't understand why uniform convergence is necessary. I figured that since the integral is linear this is trivial. I was wondering if someone could explain this to me. Maybe elaborate on what can break down, so that they aren't equal if [itex]\sum b_{j}z^{j}[/itex] doesn't uniformly converge to [itex]f(z)[/itex]

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# Uniform Convergence

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