I'm trying to show that a function [itex]f(z)[/itex] is analytic by showing [itex]f'(z)[/itex] exists. But [itex]f(z)[/itex] is defined in terms of a contour integral:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

f(z) = \oint_{|\zeta - z_0| = r} g(z,\zeta) d\zeta.

[/tex]

Since the integral is being carried out with respect to [itex]\zeta[/itex] and not [itex]z[/itex], am I allowed to bring the [itex]d/dz[/itex] operator inside the integral? Or is it more complicated than that? Are there certain conditions that [itex]g(z,\zeta)[/itex] must satisfy? If so, what are they?

THANKS!

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# Bringing a derivative inside an integral?

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