TITLE: Who introduced the discrete Fourier transform?
QUESTION [8 upvotes]: I am trying to find the original reference which introduced the definition of discrete Fourier transform as used today. When did this modern formulation (which includes the indexing from n to N-1) of DFT appear in the literature?
$$
\begin{aligned}
X_{k} &=\sum_{n=0}^{N-1} x_{n} \ e^{-\frac{i 2 \pi}{N} k n} \\
&=\sum_{n=0}^{N-1} x_{n} \left[\cos \left(\frac{2 \pi}{N} k n\right)-i \ \sin \left(\frac{2 \pi}{N} k n\right)\right]
\end{aligned}
$$
Fourier original on Analytical Theory of Heat does not deal with discrete versions. History related articles credit Gauss well before Fourier. For example here, Gauss and the History of the Fast Fourier Transform, Archive for History of Exact Sciences , 1985, Vol. 34, No. 3 (1985), pp. 265-277 ([Link])1, shows a table, but the article just defines the DFT in "modern notation" for Gauss's Latin work and credits Gauss rather.

REPLY [10 votes]: You can go earlier than Gauss if you allow for a DFT involving only sines or only cosines: I quote from Gauss and the history of the fast Fourier transform

Alexis-Claude Clairaut (1713-1765) published in 1754 what we currently
believe to be the earliest explicit formula for the DFT (the
computation for series coefficients from equally spaced samples of the
function), but it was restricted to a cosine Fourier series. Joseph Louis Lagrange (1736-1813 published a
DFT-like formula for finite Fourier series containing only sines, in
1759 and in 1762.
The earliest explicit DFT formula containing both sines and cosines
is due to Carl Friedrich Gauss (1777-1855) in "Theoria
Interpolationis Methodo Nova Tractata". It was published only
posthumously in 1866 [10], but was originally written, most likely, in
1805.

Here is Gauss's formula ( source)