References about Fourier series of a function of complex variable

Question

Hello,

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

Thanks

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.