Fourier Series are Taylor Expansions

While listening to some lectures by Richard Borcherds on modular forms, he repeatedly referred to the Taylor coefficients of modular forms as its Fourier coefficients. The first few times I thought this was just a slip of the tongue, before I realised there was actually something going on. I will thus record my thoughts here, although this is probably well-known theory which I suspect can be found in any book on Fourier analysis.

Let $f\colon [0,1]\to\mathbb C$ be a nice function with $f(0)=f(1)$. Clearly, this defines a function $S^1\to\mathbb C$, and if we consider the standard embedding $S^1\hookrightarrow\mathbb C$, we have a function of a complex variable which we denote by $g\colon S^1\to\mathbb C$ to avoid confusion. For the next paragraph, let us assume all series converge nicely and all functions behave nicely, before we start worrying about the details.

We have a complex function defined on $S^1$. If $g$ is nice, then we can extend it uniquely to an analytic function on the closed unit disk, with values defined by the Cauchy integral formula. We may then write $g$ into a power series $g(z)=\sum a_nz^n$, so that for $z=\exp(2\pi ix)$ on the unit circle, we have

$$ g(z) = \sum_{n\geq 0} a_nz^n = \sum_{n\geq 0} a_n\exp(2\pi i nx). $$

Hence, the Taylor coefficients of the analytically continued $g$ is precisely the Fourier coefficients of the original real function $f$!

The remaining question of interest is under what conditions such an analytic continuation is possible. What if we replace analytic by meromorphic and allow poles in the interior of the closed unit disk? That’s interesting to think about, but I have no great insights here as yet, so I’ll leave it there for now.