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Hello,

Is anyone who know a simple reference to discover what is Fourier series for periodic function of one complex variable ?

Thanks

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How is this a research question? –  Igor Rivin Jul 25 '11 at 21:11
    
It isn't really, but it's easy to answer definitively, and be done. –  paul garrett Jul 26 '11 at 1:36
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1 Answer 1

I am assuming that the question means something like that we have a holomorphic function $f$ that is periodic in the real part of the complex variable $z=x+iy$.

Writing a Fourier expansion in the "periodic" variable $x$, at first we only have $f(x+iy)=\sum_{n\in\mathbb Z} c_n(y) e^{2\pi in x}$, that is, that the Fourier coefficients may depend on $y$. However, from the Cauchy-Riemann equations, we find $c_n'+2\pi n c_n=0$, so $c_n(y)=C_n\cdot e^{-2\pi ny}$ for some constant. Thus, $f(z)=\sum_n C_n e^{2\pi i nz}$.

I hope this is addressing the question as intended.

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