After proving a theorem on the existance of the laurent series for a holomorphic function on an annulus, Serge Lang proceeds to further prove that they are unique. However, I do not understand his proof. Here is the theorem for the existance and the subsequent proof for their uniqueness:
In doing the integral I get that described summation integral equals $2\pi k s^k a_k$. However, I do not see how this, or even $int \sum a_n s^n e^{in \theta}$ is connected to what we are trying prove. Could someone explain to me the connection between the integral of the sum and the uniqueness of the laurent series? Any help is greatly appriciated.