Complex Analysis Homework: Proving Equation w/ Cauchy's Integral Formula

[asi123]

Homework Statement

Hey guys.
Look at this question please.
I have two paths, and I need to proof the thing in red.
They give a tip due, they say to show first that the equation in green is correct and then using Taylor development, to proof the red equation.
I'm still in the first phase, in the green equation. I know it got something to do with Cauchy's integral formula but I can't see how...
Any idea guys?

Thanks a lot.

Homework Equations

The Attempt at a Solution

 

[lippyka]

If I understand the question correctly, then you are trying to prove that the equation in green is correct using Cauchy's Integral Formula. To do this, you first need to understand what Cauchy's Integral Formula is and how it works. Cauchy's Integral Formula states that if a complex function f(z) has a continuous derivative on a simple closed contour C, then the integral of the function around the path will be equal to 2πi times the sum of all the residues of the function inside the contour. That is, \int_C f(z) \ dz = 2 \pi i \sum_{z_0 \in C} Res[f,z_0] Now let's take a look at the equation in green. It says that the integral of the complex function f(z) along the contour C is equal to the integral of its derivative, f'(z), along the same contour. This can be written as \int_C f(z) \ dz = \int_C f'(z) \ dz Using Cauchy's Integral Formula, we can rewrite this as 2 \pi i \sum_{z_0 \in C} Res[f,z_0] = 2 \pi i \sum_{z_0 \in C} Res[f',z_0] Which is equivalent to saying that the sum of the residues of f(z) is equal to the sum of the residues of f'(z). This is true for any simple closed contour C, so the equation in green holds.