MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top


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


share|cite|improve this question
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

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.