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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

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

In the book Fourier Series and Wavelets, J.P. Kahane, P.G. Lemarié-Rieusset, Gordon and Breach Publishers, 1995, pp. 1, the authors states that

In the last paragraph of page 2, they wrote

"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 is given to this statement.

ADDED: KAHANE, J.-P.; LEMARIÉ-RIEUSSET, P.G., Fourier series and wavelets Part I: Fourier series, is available as a Publications Mathématiques d'Orsay here. Part II of this work is available here.

After a insane search :-) I have found a paper from René Cazenave, Ingenieur a la Societe d'Electronique et d'Automatisme, "Convergence et sommation d’une série de Fourier correspondant a une fonction analytique", published on Annals of TelecommunicationsVolume 10, Number 5, 102-108, DOI: 10.1007/BF03013655, dated May, 1955. I have no access to the complete paper. I have access to a pdf-preview from the first page of this paper (http://http://www.springerlink.com/content/827383p763vx7j45/).

The first section of this paper, named "Connexion des Séries de Fourier et des Fonctions Holomorphes d'une Variable Complex" presents the Complex Fourier Series, equation (4).

The author cites a reference: GOURSAT, E., Cours d'analyse mathematique, 5ed, 1942, Tome III, Nos. 509 et 518. Unfortunatelly, I have no access to this specific book edition too. This book may be the key to answer the question!

ADDED: There is a English translation for Goursat's book. This translation is available here but I have not found in it anything relevant to the subject of the question.

By the way, I will continue to search :-)

-- 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 full paper so I can not search for references.

Zygmund, A., Trigonometrical Series, 1935, chap.1, pp.2: "1.18 The complex form of trigonometrical series. Applying Euler's formulae to coskx, sinkx ...."

The paper begins with a very beautiful equation:

" Considérons la série

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

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

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.

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

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

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

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.

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

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

In the book Fourier Series and Wavelets, J.P. Kahane, P.G. Lemarié-Rieusset, Gordon and Breach Publishers, 1995, pp. 1, the authors states that

In the last paragraph of page 2, they wrote

"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 is given to this statement.

ADDED: KAHANE, J.-P.; LEMARIÉ-RIEUSSET, P.G., Fourier series and wavelets Part I: Fourier series, is available as a Publications Mathématiques d'Orsay here. Part II of this work is available here.

After a insane search :-) I have found a paper from René Cazenave, Ingenieur a la Societe d'Electronique et d'Automatisme, "Convergence et sommation d’une série de Fourier correspondant a une fonction analytique", published on Annals of TelecommunicationsVolume 10, Number 5, 102-108, DOI: 10.1007/BF03013655, dated May, 1955. I have no access to the complete paper. I have access to a pdf-preview from the first page of this paper (http://http://www.springerlink.com/content/827383p763vx7j45/).

The first section of this paper, named "Connexion des Séries de Fourier et des Fonctions Holomorphes d'une Variable Complex" presents the Complex Fourier Series, equation (4).

The author cites a reference: GOURSAT, E., Cours d'analyse mathematique, 5ed, 1942, Tome III, Nos. 509 et 518. Unfortunatelly, I have no access to this specific book edition too. This book may be the key to answer the question!

ADDED: There is a English translation for Goursat's book. This translation is available here but I have not found in it anything relevant to the subject of the question.

By the way, I will continue to search :-)

-- 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 full paper so I can not search for references.

Zygmund, A., Trigonometrical Series, 1935, chap.1, pp.2: "1.18 The complex form of trigonometrical series. Applying Euler's formulae to coskx, sinkx ...."

The paper begins with a very beautiful equation:

" Considérons la série

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