Theorem 12.1 (Fourier Expansion). Assume that [Graphics:Images/FourierSeriesComplexMod_gr_95.gif] is the Fourier Series for [Graphics:Images/FourierSeriesComplexMod_gr_96.gif]. If [Graphics:Images/FourierSeriesComplexMod_gr_97.gif] are piecewise continuous on [Graphics:Images/FourierSeriesComplexMod_gr_98.gif], then [Graphics:Images/FourierSeriesComplexMod_gr_99.gif] is convergent for all [Graphics:Images/FourierSeriesComplexMod_gr_100.gif].  

The relation [Graphics:Images/FourierSeriesComplexMod_gr_101.gif] holds for all [Graphics:Images/FourierSeriesComplexMod_gr_102.gif]where U is continuous. If [Graphics:Images/FourierSeriesComplexMod_gr_103.gif] is a point of discontinuity of U, then

         [Graphics:Images/FourierSeriesComplexMod_gr_104.gif],  

where [Graphics:Images/FourierSeriesComplexMod_gr_105.gif] denote the left-hand and right-hand limits, respectively.  With this understanding, we have the Fourier Series expansion:

        [Graphics:Images/FourierSeriesComplexMod_gr_106.gif] .  

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