Theorem 12.1 (Fourier Expansion). Assume that is the Fourier Series for . If are piecewise continuous on , then is convergent for all .  

The relation holds for all where U is continuous. If is a point of discontinuity of U, then

         ,  

where denote the left-hand and right-hand limits, respectively.  With this understanding, we have the Fourier Series expansion:

         .  

Proof.

12.1.1 Proof of Euler's Formulas.

    A complete discussion of the details of the proof of Theorem 12.1 can be found in some advanced texts.  See for example, John W. Dettman, Chapter 8 in "Applied Complex Variables", The Macmillan Company, New York, 1965.

Theorem 12.1 is mentioned in the book.

Complex Analysis for Mathematics and Engineering

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