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I am reading Carlslaw's "Introduction to the Theory of Fourier's Series and Integrals", first chapter on the history of Fourier series, page 3. The author asserts that Clairaut and Euler did derive in parts the coefficients of Fourier series. He said Euler wrote:

$$f(x)=a_0+2a_1\cos(x)+2a_2\cos(2x)+...2a_n(\cos(n)$$

Euler multipled both sides $\cos(nx)$ and integrated the series from $0$ to $\pi$ term by term to obtain:

$$a_n=\int_{0}^{\pi}f(x)\cos(nx)dx$$

His reference is Petrop.N.Acta, 11, 1793(1798), page 94 (May 1777).

I checked it and find "Methodus facilis inveniendi series per sinus sosinusve angulorum multiplorum procedentes, quarum usus in universa theoria astronomiae est amplissimus". I think this is the correct paper, but I cannot find the reference that the author made.

In that article, I only see on page 101, this:

$$S=1+\cos(x)+\cos(2x)+\cos(3x)+...$$

Multiply by $2\sin(\frac{x}{2})$

One get:

$$2\sin(\frac{x}{2})S=2\sin(\frac{x}{2})+2\sin(\frac{x}{2})\cos(x)+2\sin(\frac{x}{2})\cos(2x)+2\sin(\frac{x}{2})\cos(3x)+...$$

$$2\sin(\frac{x}{2})S=2\sin(\frac{x}{2})+\sin(\frac{3}{2}x)-\sin(\frac{3}{2}x)+\sin(\frac{5}{2}x)-\sin(\frac{5}{2}x)...+\sin(n+\frac{1}{2}x)-\sin(n-\frac{1}{2}x)$$

This is on page 101.

I couldn't find the reference for this integral, can you help me to find it?

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1 Answer 1

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Euler's calculation of Fourier series coefficients appears in his paper De serierum determinatione seu nova methodus inveniendi terminos generales serierum from 1753. A link to it on the Euler archive is https://scholarlycommons.pacific.edu/euler-works/189/: see the second pdf link at the bottom, on page 81, which is the end of the solution to Problem IX just above Corollary I.

A cleaner scan from the Opera Omnia (Series I, volume 14, p. 511) is on the Hathitrust website https://babel.hathitrust.org/cgi/pt?id=pst.000010974780&view=1up&seq=527 (this link may not work everywhere). See the equation just before Corollary 1 on that page.

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