Theorem 12.5  (Termwise Integration).  Assume that    has the Fourier series representation  

          .  

Then the integral  of    has a Fourier series representation which can be obtained by termwise integration of the Fourier series of  ,  that is  

        ,  

where we have used the expansion  .  

Proof.

Proof of Theorem 12.5 is left for the reader in the book.

Complex Analysis for Mathematics and Engineering

(c) 2006 John H. Mathews, Russell W. Howell