[Math] complex fourier coefficients, introduced by

Question

I remember reading somewhere that the complex Fourier coefficients were introduced by an engineer sometime around 1900, but I can't find anymore this information.

Does anyone know the name of this person and where I can find a reference to it?

EDIT: I state the question more clearly: "Who was it that first wrote a Fourier series not as a sum of sines and cosines but as a sums of complex exponentials, with the relative formula for the coefficients?". I may be totally wrong about this all, since I don't remember well and that's why I'm asking. Also don't take the 1900 thing seriously, I may be off by 50+ years.

Answer 1

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