n=1 . an cos L N=0 . M n=1 nic Problem 6.2. i) Using the Parseval's identity for generalized Fourier series corresponding to delo in (0,L) with a (0)(L)=0 to derive *g2 (x)dx = La + Zaz for any pw 2 smooth function g with g(x) = being its Fourier cosine series in (0,L). ii) For smooth f defined on (0, L) with f() = f(1) = 0, let f(x)=bn sin( - be its Fourier L sine series and f(x)= bn sin( -) be the first M term truncation, using the orthogonality of L L the basis (sin("")} to establish that - sin(' dic 6. 2 iii) For f(x) in ii) and g(x) = f'(x), using term-by-term differentiation and the results of i) & ii) L2 to show f(x) - om sin L 72(M + 1)2 (This shows how quickly the Fourier sine series converges, if f has square integrable ist derivative.) n=1 S." |sca) - nic L n=M+1 2 nic force |() o das 2(M + 1); **()?de= ${r'(=)?de . n=1 n=1 . an cos L N=0 . M n=1 nic Problem 6.2. i) Using the Parseval's identity for generalized Fourier series corresponding to delo in (0,L) with a (0)(L)=0 to derive *g2 (x)dx = La + Zaz for any pw 2 smooth function g with g(x) = being its Fourier cosine series in (0,L). ii) For smooth f defined on (0, L) with f() = f(1) = 0, let f(x)=bn sin( - be its Fourier L sine series and f(x)= bn sin( -) be the first M term truncation, using the orthogonality of L L the basis (sin("")} to establish that - sin(' dic 6. 2 iii) For f(x) in ii) and g(x) = f'(x), using term-by-term differentiation and the results of i) & ii) L2 to show f(x) - om sin L 72(M + 1)2 (This shows how quickly the Fourier sine series converges, if f has square integrable ist derivative.) n=1 S." |sca) - nic L n=M+1 2 nic force |() o das 2(M + 1); **()?de= ${r'(=)?de . n=1