Theorem 12.4 (Fourier Sine Series). Assume that [Graphics:Images/FourierSeriesComplexMod_gr_199.gif] is an odd function and has period [Graphics:Images/FourierSeriesComplexMod_gr_200.gif]. If [Graphics:Images/FourierSeriesComplexMod_gr_201.gif] are piecewise continuous, the Fourier series for   [Graphics:Images/FourierSeriesComplexMod_gr_202.gif]  involves only the sine terms,  [Graphics:Images/FourierSeriesComplexMod_gr_203.gif],  and we write  

        [Graphics:Images/FourierSeriesComplexMod_gr_204.gif],  

where  [Graphics:Images/FourierSeriesComplexMod_gr_205.gif].  

Proof.

Proof of Theorem 12.4 is left for the reader in the book.

Complex Analysis for Mathematics and Engineering

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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