Using non-integer inputs for the Fourier Series

Question

I am a beginner in the topic of the Fourier Series. I have been doing some reading and, according to this website (https://www.thefouriertransform.com/series/complexcoefficients.php), the following function "cn"1 (which belongs to the Fourier Series) takes in "n" as an input, and n has to be an integer. My question is: why does it have to be an integer, and not any number at all?

Answer 1

The discussion here is about Fourier series. A Fourier series represents a periodic function $g : \Bbb R\to \Bbb R$ with period $T$ as an infinite sum $$g(t) = \sum_{n = -\infty}^\infty c_ne^{i\frac{2\pi}Tnt}$$ $c_n$ is not a function named "cn". Rather, it is the $n$-th term of a sequence labelled with the letter $c$:$$(\dots, c_{-3}, c_{-2}, c_{-1}, c_0, c_1, c_2, c_3,\dots)$$ Technically $c$ is a function from the integers $\Bbb Z \to \Bbb C$ and $c_n$ is another way of denoting $c(n)$, the value of the function for the input $n$, but generally we think of sequences as collections of numbers ordered in a certain way.

This scheme does not quite require that $n$ is limited to integer values. It could be any collection of discrete values. Discrete meaning that each allowed $n$ will have some distance around it in which no other index values lie. But there is no reason to use some other set than the integers for this. The same summation can be obtained using integers, so anything else is just making it more complicated for no advantage. It is the summation that requires the index have discrete values. The summation is defined by summing finite collections of the values, getting partial sums, and taking limits to extend it to all of $\Bbb Z$. The summing of finite sets of values is automatically discrete, and the only limits involved here are about reaching the unbounded ends, not closing the gaps between indicesThis discrete summation is what it means to be a "Fourier series".

Now you can replace that summation with an integration over a continuous variable: $$g(t) = \int_{-\infty}^{\infty}c(x)e^{i2\pi xt}dx$$ This is the Fourier Transform, related to but not quite the same thing as the Fourier series. (Actually, it is the Inverse Fourier Transform, as the transform itself finds the function $c$ needed to represent $g$, while the inverse transform reconstructs $g$ from $c$, as seen here.) The period $T$ has disappeared because the Fourier transform can represent non-periodic functions $g$. In fact, the only way to get a periodic $g$ is to use more general concepts of integration than Riemann's. And even then it just devolves into the Fourier series instead.