Fourier Series

Figure 1.4.4. Extension of half-range cosine series.The function is extended to be periodic as an even function in −, andisshowninfigure 1.4.4; the Fourier coefficients are given bya 0 2 0/2xdx 2 /2x − 2dx 2a n 2 0/2xcosnx dx 2 /2x − 2cosnx dx 1 n sin n 21 − 4 n sin n 2Now sinn/2 is non-zero (zero) when n is odd (even), so we write n 2m − 1toretainonly the non-zero terms and start our series with m 1; this givesa 2m−1 12m − 12m − 1sin21 − 42m − 1 sin2m − 1 2.Nowsin2m − 1 2 1 m 1,3,5.... −1 m 2,4,6... −1 m−1Thus the required series isfx ~ 4 ∑m1−1 m−12m − 11 4−1 m2m − 1cosnx28