sikrut
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The sine-cosine (SC) Fourier series: $$f(x) = \frac{A_0}{2} + \sum_{j=1}^{+\infty} A_j cos(jx) + \sum_{j=1}^{+\infty} B_jsin(jx) $$
This form can also be expanded into a complex exponential (CE) Fourier series of the form: $$ f(x) = \sum_{n=-\infty}^{+\infty} C_n e^{inx} $$
and vice versa. Here we define, for convenience: $$ B_n := 0 $$
(a) Prove that the SC Fourier coefficients ##A_j## and ##B_j## for non-negative integer ## j = 0,1,2,3...## are related to the CE Fourier coefficients ##C_n## by: $$ A_j = C_j + C_{-j} ; \\ B_j = i(C_j - C_{-j}) ; \\ for: j = 0,1,2,3... $$
Hint: All you need here is Euler: ## e^{i\gamma} = cos(\gamma) + isin(\gamma) ## or ## cos(\gamma) = (e^{i\gamma} + e^{-i\gamma})/2 ## or ## sin(\gamma) = (e^{i\gamma} - e^{-i\gamma})/(2i) ##
I plugged in the Euler formula to the CE Fourier series and evaluated the definite integral from -pi to pi, which equaled zero, soooo I'm not too sure what to do...
This form can also be expanded into a complex exponential (CE) Fourier series of the form: $$ f(x) = \sum_{n=-\infty}^{+\infty} C_n e^{inx} $$
and vice versa. Here we define, for convenience: $$ B_n := 0 $$
(a) Prove that the SC Fourier coefficients ##A_j## and ##B_j## for non-negative integer ## j = 0,1,2,3...## are related to the CE Fourier coefficients ##C_n## by: $$ A_j = C_j + C_{-j} ; \\ B_j = i(C_j - C_{-j}) ; \\ for: j = 0,1,2,3... $$
Hint: All you need here is Euler: ## e^{i\gamma} = cos(\gamma) + isin(\gamma) ## or ## cos(\gamma) = (e^{i\gamma} + e^{-i\gamma})/2 ## or ## sin(\gamma) = (e^{i\gamma} - e^{-i\gamma})/(2i) ##
I plugged in the Euler formula to the CE Fourier series and evaluated the definite integral from -pi to pi, which equaled zero, soooo I'm not too sure what to do...