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Bringing a derivative inside an integral?

  1. Oct 30, 2009 #1
    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:
    [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!
     
  2. jcsd
  3. Oct 30, 2009 #2

    Gib Z

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  4. Nov 6, 2009 #3
    I think you might want to use leibnitz rule.
     
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