Solving the Integral in Proving Analyticity of g(z)

[laura_a]

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!

 

[siddharth]

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?