In the book *Fourier Series and Wavelets*, J.P. Kahane, P.G. Lemarié-Rieusset, Gordon and Breach Publishers, 1995, pp. 1 (available as a *Publications Mathématiques d'Orsay* here), the authors state that

"The subject matter of Fourier series consists essentially of two formulas :

(1) $$f(x) = \sum c_n e^{inx}, $$

(2) $$c_n = \int f(x) e ^{-inx} \frac{dx}{2 \pi}.$$

The first involves a series and the second an integral."

In the last paragraph of page 2, they add: "It is time to say that formulas (1) and (2) were never written by Fourier. **Complex exponentials were not used in Fourier Series until well into the twentieth century**".

Unfortunately, no references are given to this statement.

-- UPDATE 1: **1935**

G.H.Hardy, J.E.Littlewood, "Notes on the theory of series (XIX): A problem concerning majorants of fourier series", Quartely Journal of Mathematics, Vol os-6, Issue 1, pp. 304-315, 1935, equation (1.1.1) explicit complex Fourier series equation. I have no access to the full paper so I cannot search for references.

A. Zygmund, Trigonometrical Series, 1935: §§1.13 (p. 2) and 1.43 (p. 6) give formulas (1) and (2) above.

-- UPDATE 2: **1892**

J. de Séguier, "Sur la série de Fourier", Nouvelles annales de mathématiques 3e série, tome 11, p. 299-301, 1892

The paper begins with a very beautiful equation: "Considérons la série

$$S = \sum_{n =-\infty}^{+\infty} \frac{e^{\frac{2n i \pi z}{\omega}}}{\omega} \int_{u_0}^{u_0 + \omega} f(u)
e^{-\frac{2n i \pi u}{\omega}}du.\text{"} $$

As we can see, the integral part of this equation is the complex Fourier coefficients. Therefore, $S$ represents the complex Fourier series.

One interesting conclusion: **By the equation published in this paper, complex exponentials were used in Fourier Series BEFORE twentieth century**

-- UPDATE 3: **1875**

M.M. Briot et Bouquet, Théorie des Fonctions Elliptiques, Deuxième Édition, Gauthien-Villars, Paris, 1875

Under the title "Série de Fourier" (page 161), at page 162 we can see equations (2) and (3) that are expressions of Fourier series with complex exponentials.

The link for page 162: http://gallica.bnf.fr/ark:/12148/bpt6k99571w/f172.image

Traité de la chaleu, and I have to say I am quite surprised that I did not find any trigonometric series written in the exponential form. He is of course well aware of Euler's formula---he uses it several times---but he seems not to be moved to write Fourier series using it. It would be simply extraordinary if this had had to wait for the 1900s though! $\endgroup$ – Mariano Suárez-Álvarez Mar 13 '11 at 21:553more comments