# Cauchy Integral formula

Hello:

I am wondering about the following:

Let C be the circle |z| = 3. the contour integral

g(w) = integral on C of (2*z^2-z-2/z-w)dz can be evaluated by cauchy's integral formula. I am wondering what happens if w is greater than 3.

Would you get this: f = 2*z^2-z-2. This is an entire function.

because the contour integral of an analytic function f around 2 simply connected contours is equal, so for a contour C2 that is greater than |z| = 3, the integral of f over that contour would be equal to the integral of f over contour C. So, this means that g(w) would still give you the value of the contour integral even though your contour covered a greater area than |z| = 3. So, this would mean that for w > 3, you can still use the Cauchy formula and get 2*pi*i*2*w^2-w-2? Thanks very much.

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Dick
Homework Helper
The integral around two contours are the same only if they enclose the same poles and wind in the same direction. If |w|>3 then its not in your circular contour. If another contour does enclose the pole then its value is not the same as the circular contour.

Cauchy's Formula

Hello,

but sorry, could you give me some hints as to how to do the problem then?

I guess what i mean is, why can't i draw a big circle that covers all |w| >3 and still use the same integral formula? Thanks.

Dick
Homework Helper
Because you want the integral over the circle C, not the integral over the bigger contour. You'll have to divide the problem into two cases. |w|<3 and |w|>3.

Oh, okay. so would the value be zero then? since the function, if |w| >3, is analytic throughout C (|z| = 3), and the integral of an analytic function over a simply connected closed contour = 0 if f is analytic everywhere inside and on the boundaries of the contour? Thanks.

Dick