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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.

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I'm not sure I understand the question. Is the question something like: when did the Fourier representation of a (say real-valued) function in terms of complex exponentials come to predominate over the representation in terms of sines and cosines? I am having trouble believing it was really an engineer in 1900 who first introduced this, but it would be interesting to know more of the history, and how it might have led to the introduction of inner product spaces, Hilbert space, etc. (I'm also conflicted as to the appropriateness of this question for MO. Anyway, please clarify the question.) –  Todd Trimble Mar 13 '11 at 14:29
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I ave just spent a few minutes browsing Fourier's 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! –  Mariano Suárez-Alvarez Mar 13 '11 at 21:55
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(One can get scans of the book from archive.org; it is amazingly readable!) –  Mariano Suárez-Alvarez Mar 13 '11 at 22:04
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The Riesz--Fischer theorem (1907) on the L^2-properties of Fourier series is a result where the complex exponential form of Fourier series is an essential ingredient. I don't have access to their paper, but on account of it I would guess that a systematic use of complex exponentials in Fourier series came no later than 1907. –  KConrad Mar 13 '11 at 22:29
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Actually, maybe there are no complex Fourier series in the papers by Riesz or Fischer; one could imagine their theorem being about L^2-spaces of real-valued functions. So someone should check their papers to see what's in them. –  KConrad Mar 13 '11 at 23:01
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4 Answers

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

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Here's a link for the paper by Séguier: archive.numdam.org/article/NAM_1892_3_11__299_1.pdf –  Barry Cipra Apr 10 '12 at 0:16
    
I cannot comment because I don't have enough reputation, but here is the remark: In Zygmund's classical treatise on Fourier series, the complex exponential is not used. His book was published in the 50's. Thus, Kahane is (obviously) right, the complex representation did not come to prevail well until the 20-th century. I find it curious myself. –  kolik May 27 '12 at 23:40
    
The first equation (equation (1), pp. 3) of "Location of Best Approximation", S. Bochner, published in Zygmund's "Contributions to Fourier Analysis", Princeton Univ. Press, 1950, IS the complex Fourier serie... (see archive.org/stream/contributionstof030473mbp#page/n13/mode/2up) Are you referring to this book? –  Papiro May 28 '12 at 12:08
    
@PaPiro: I think kolik means ams.org/mathscinet-getitem?mr=0107776 . However that book does -- like the first edition you quoted before -- contain the complex form: eqs (4.2) and (4.3), pp. 6-7. –  Francois Ziegler Jun 1 '12 at 4:37
    
To be precise it is correct that Zygmund mentions the complex representation. However for the bigger part of the book it is the sin, cos notation which is used. –  kolik Jun 1 '12 at 15:25
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An "engineer" playing with Fourier series around 1900 sounds to me like Oliver Heaviside. In Electromagnetic Theory volume 2, chapter 279, equation 112 (dating from 1895) there is a complex representation of a Fourier series. This is might be the origin of what you read, from the previous answer it might not, however, be the first instance.

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"The subject of the decomposition of an arbitrary function into the sum of functions of special types has many fascinations. No student of mathematical physics, if he possess any soul at all, can fail to recognize the poetry that pervades this branch of mathematics. The great work of Fourier is full of it, although there only the mere fringe of the subject is reached. For that very reason, and because the solutions can be fully realised, the poetry is more plainly evident in cases of greater complexity" Oliver Heaviside, On the self-induction of wires, Part III, p.201, Electrical Papers,1894 –  Papiro May 10 '12 at 15:35
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I believe the answer to the question in the title is probably Cauchy, who in Méthode simple et générale pour la détermination numérique des coefficients que renferme le développement de la fonction perturbatrice, C. R. Acad. Sci. Paris 11 (1840), 453-475, writes (page 469):

"La formule $$ Q = \sum Q_{h,h'} e^{h(p-\varpi)\sqrt{-1}} e^{h'(p'-\varpi')\sqrt{-1}} $$ entraîne l'équation $$ Q_{h,h'} = \frac1{4\pi^2} \int_0^{2\pi} \int_0^{2\pi} Q e^{-h(p-\varpi)\sqrt{-1}} e^{-h'(p'-\varpi')\sqrt{-1}} dp\\,dp'." $$ Specializing to $\varpi=\varpi'=0$ one gets the requested complex Fourier series and formula for its coefficients.

[Edit: H. Burkhardt in Trigonometrische Reihen und Integrale, Encykl. Math. Wiss. II A 12, p. 929 confirms the above as the first of several papers where Cauchy uses the complex form. Moreover he goes back even further to Laplace who writes in Théorie analytique des probabilités (Paris, 1812), pp. 83-84:

"Take the equation $u=\sum_{x=0}^\infty y_xt^x$. Substitute on both sides $e^{x\varpi\sqrt{-1}}$ for $t^x$... and write $U$ for what $u$ then becomes. Multiplying the equation by $e^{-x\varpi\sqrt{-1}}$ and integrating... the right-hand side boils down to $2\pi y_x$; one has therefore $y_x = \frac1{2\pi}\int U\ d\varpi\ (\cos x\varpi - \sqrt{-1}\sin x\varpi)$".]

Note also that specific complex Fourier series were written much earlier. For instance Lagrange in 1766 computes $$ (1-\alpha e^{i\theta})^{-s} = \sum_{m=0}^{\infty}\binom{-s}{m}(-\alpha)^me^{im\theta} $$ ... except that he still writes everywhere $e^{im\theta}$ in the form $\cos m\theta +\sin m\theta\sqrt{-1}$.

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As another data point, my copy of Whitaker and Watson A course of Modern Analysis, 4th edition from 1927 uses the trig versions for basically everything, but Example 1, $$f(z) = \sin(z) - 1/2 \sin 2z + 1/3 \sin(3x) -\ldots$$ is immediately converted to $$f(z) = \frac{1}{2}i(e^{iz}-1/2e^{2iz}+\ldots)+\frac{1}{2}i(e^{-iz}-1/2e^{2iz}+\ldots)$$ The references given "for a fuller account of investigations subsequent to Riemann" are Hobson's Functions of a Real Variable and de la Vallée Poussin's Cours d'Analyse Infinitésimale.

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