Solving the Integral in Proving Analyticity of g(z)

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SUMMARY

The discussion centers on proving that the function g(z) = 1/(2πi) ∫_C f(s) ds/(s-z) is analytic, given that f is continuous on a simple closed contour C. Participants highlight the necessity of applying the Cauchy Integral Formula and the Cauchy-Riemann equations to establish analyticity. A key point raised is the challenge of differentiating a function that contains an integral, prompting inquiries about the appropriate version of the Cauchy Integral Formula to use in this context.

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Homework Statement



My textbook has this question

f is continuous on C which is a simple closed contour...

Prove g(z) = 1/(2*pi*i) int_c f(s)ds/(s-z) is analytic?

I understand that you use Cauchy-Riemann proof for analytic functions, but how do I find the derivatve of a function that has an integral inside of it? Any suggestions?



Homework Equations



I thought maybe somehow I'd have to use the Cauchy Integral formula, but what version of it should I use? and should I even use that - the only reason I think that is that is the question in the textbook after the Cauchy Integral topic! :)

The Attempt at a Solution



I'm not sure where to start because I don't know how to differentiate the funciton because of the integral there and not sure how to get rid of it!
 
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laura_a said:

f is continuous on C which is a simple closed contour...
Prove g(z) = 1/(2*pi*i) int_c f(s)ds/(s-z) is analytic?


Is that exactly how the question is stated? Does it say that f is continuous on C, or does it say f is analytic on and inside C?
 

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