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Why in the Fourier series for $e^x,A_n=C_n+C_{-n}$ and $B_n=i(C_n+C_{-n})$, where $A_n,B_n$ are the coefficients of full Fourier series and $C_n$ is the coefficient of full Fourier series in complex form ?

Does this always holds, i.e. the coefficients of the full Fourier series can be obtained from $C_n$ ? A yes or no it's enough as an answer (I'll do the rest).

I already tried to add both coefficients I get

$A_n+iB_n=lC_n$.

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Hint

You have for $n \geq 0$: $$C_n e^{i n x}+C_{-n} e^{-i n x}=C_n \left( \cos(nx)+i\sin(nx) \right)+C_{-n} \left( \cos(nx)-i\sin(nx) \right)=\left( C_n +C_{-n} \right) \cos(nx)+i \left( C_n-C_{-n} \right) \sin(nx)$$ you can then sum over $n$.

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  • $\begingroup$ As you give a hint I assume $B_n=i(C_n+C_{-n})$ and $A_n=(C_n+C_{-n})$ always hold? $\endgroup$ – MagicConchShell Apr 29 '18 at 3:13
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    $\begingroup$ @Isa Yes but it seems that $B_n=i(C_n \color{blue}{-}C_{-n})$ :-) $\endgroup$ – Delta-u Apr 29 '18 at 8:28

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