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0 $begingroup$ i’d like to calculate Fourier coefficients of $cos 2 pi f_0 t$ . This is what I did : $$ c_k = frac{1}{T_0}int_{0}^{T} cos 2 pi f_0 t cdot e^{-2ipi f_0 t}. $$ From Euler formulas: $$ frac{1}{T_0}int_{0}^{T} frac{ e^{2ipi f_0 t} + e^{-2ipi f_0 t} }{2} cdot e^{-2ipi f_0 t}. $$ $ frac{1}{2T_0}int_{0}^{T} 1 + e^{-2pi f_0 t(1+i) } $ Solving i obtained $c_k = frac{1}{2T_0}[T_0 + frac{e^{-2pi (1+i)} - 1 }{-2pi f_0 - 2 i pi f_0 }] $ (because $f_0 = 1/T_0 $ ). But $e^{-2ipi}e^{-2pi} = e^{-2pi}[cos2pi + i sin(-2pi)]= e^{-2pi}$ . So $frac{1}{2T_0}[ T_0 + frac{e^{-2pi } - 1 }{-2pi f_0}(1+i)]$ . On my book the result is $1/2$ If $k = pm 1$ And $0$ If $k$ Is different from $1$ . Can someone can tell me where is the error ? I’d really like to learn how to correct this exercise . Thank ...