Homework 2 collaboration area
Here it is again:
$ f(a)=\frac{1}{2\pi i}\int_\gamma \frac{f(z)}{z-a}\ dz. $
This is the place.
Exercise 1
Suppose that $ \varphi(z) $ is a continuous function on the trace of a path $ \gamma $. Prove that the function
$ f(z)=\int_{\gamma}\frac{\varphi(\zeta)}{\zeta-z}d\zeta $
is analytic on $ \mathbb{C}-\text{tr }\gamma $.
Discussion
-A function is said to be analytic on an open set $ \Omega $ if it is $ \mathbb{C} $-differentiable at every point in $ \Omega $. Since the definition of analytic involves an open set, to complete Exercise 1 we need to have an open set somewhere. Since $ \text{tr }\gamma $ is the continuous image of a compact set, it is compact. Since $ \mathbb{C} $ is a Hausdorff space, $ \text{tr }\gamma $ is closed. Alternately one can note that $ \mathbb{C} $ is homeomorphic to $ \mathbb{R}^{2} $ where we know that a compact set is closed and bounded. Since the complement of a closed set is open, $ \mathbb{C}-\text{tr }\gamma $ is open. The long and short of this is: $ \mathbb{C}-\text{tr }\gamma $ is an open set so one just has to show that $ \varphi $ is $ \mathbb{C} $-differentiable at every point in this set in order to complete the exercise.
