# Complex Analysis: Cauchy's Theorem

• May 4th 2010, 08:01 PM
jokusan13
Complex Analysis: Cauchy's Theorem
As I am studying for an exam I am trying to wrap my head around the concepts I learned. I want to make sure I fully understand the concepts before the exam in 1.5 weeks.

Cauchy's Theorem
If u and v satisfy the Cauchy-Riemann equations inside and on the simple closed contour C, then the integral of f(z)=0

Now for example, if we have f(z)=1/(z+20) and our closed contour is a circle around the origin with radius=1. If I am understanding this correctly, we can say that the integral is equal to 0 since the 'bad point' of z=-20 is outside of the circle correct meaning that f is differentiable in and on |z|=1.

Does this sound correct?
• May 5th 2010, 02:24 AM
HallsofIvy
Quote:

Originally Posted by jokusan13
As I am studying for an exam I am trying to wrap my head around the concepts I learned. I want to make sure I fully understand the concepts before the exam in 1.5 weeks.

Cauchy's Theorem
If u and v satisfy the Cauchy-Riemann equations inside and on the simple closed contour C, then the integral of f(z)=0

Now for example, if we have f(z)=1/(z+20) and our closed contour is a circle around the origin with radius=1. If I am understanding this correctly, we can say that the integral is equal to 0 since the 'bad point' of z=-20 is outside of the circle correct meaning that f is differentiable in and on |z|=1.

Does this sound correct?

Yes, that is correct. The, around a closed path, of any function that is analytic inside and on the path is 0.