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Joseph Fourier's Théorie analytique de la chaleur by Joseph Fourier (1822) makes trigonometric series instruments for solving problems of heat conduction. A prescribed temperature distribution need not be given by one power series: it may describe different heating on different parts of a body. The extract occurs in Chapter III, §233, where the earlier assumption of an odd-power expansion is relaxed. The innovation concerns both representation and what can count as a function in an analytical calculation.^[ref]

Use a separate integration variable $t$ to unpack the printed notation. The proposed Fourier sine series has coefficientsThe integral signs in the extract denote definite integrals over $[0,\pi ]$, not antiderivatives; the author explicitly supplies these bounds in the surrounding sentence. Their absent printed bounds are a notational convention, not permission to choose arbitrary constants of integration.

The key calculation is the orthogonality relationThe product-to-sum formulas prove this by integrating cosines of integer multiples of $t$. If an expansion can be integrated term by term, multiplication by $\sin (mt)$ followed by integration selects exactly one coefficient. This derives the only possible coefficients, but does not by itself prove the expansion converges to the proposed function. That distinction is central to assessing the extract's claim.

For a revealing example take $f(x)=1$ on $0<x<\pi$. Direct integration gives $b_n=2(1-(-1{)}^n)/(\pi n)$, and thereforeEvery term is zero at $0$ and $\pi$, so the series cannot take the value $1$ at those endpoints. Its natural interpretation uses the odd extension followed by periodic extension; the endpoints become jump discontinuities. More generally, for a piecewise continuously differentiable function, the Fourier series converges at an interior jump to the average of the two one-sided limits, irrespective of an individually assigned value there. Changing one value also leaves the integrals unchanged. Consequently the unqualified assertion of representation for every arbitrary function cannot be a modern pointwise theorem. Dirichlet's theorem on Fourier series supplies sufficient conditions and the correct distinction between continuity points and jumps; its 1829 source explicitly addresses the proof problem.^[ref]

The physical payoff is explicit. For a rod scaled to $0<x<\pi$ with zero end temperature, separation of variables in the heat equation producesEach mode solves $u_{\tau }=\kappa u_{xx}$ and the end boundary conditions; the Fourier sine coefficients encode the initial temperature. This rod is an explanatory example, distinct from the stationary rectangular-solid problem opening Joseph Fourier's Chapter III. The factor $e^{-\kappa n^2\tau }$ damps high modes rapidly for positive time. The representation of arbitrary data makes the physical solution possible; the proof of that representation remains a separate task.

By Dirichlet's theorem on Fourier series, a piecewise continuously differentiable periodic function has a Fourier series converging at each point to the average of its one-sided limits. In particular it converges to the function at every point of continuity. Our periodic extension satisfies these hypotheses, and $x=\pi /2$ is a point of continuity with $f(\pi /2)={\pi }^2/4$.

At this point $\sin (n\pi /2)=0$ for even $n$. For $n=2j-1$,Substituting in the Fourier series from part (c) provesThe sum is taken in its displayed order. Discarding its zero even-indexed terms is valid and does not rearrange the remaining terms.

Using the usual piecewise smooth interpretation of the derivative hypothesis, Dirichlet's theorem on Fourier series gives the average of the two one-sided limits of the appropriate periodic extension. The even extension is continuous at $0$ and at the identified endpoints $\pm \pi$, so the Fourier cosine series converges toFor the odd extension, the one-sided limits at zero are $-f(0)$ and $f(0)$, while at $\pi$ the periodic one-sided limits are $f(\pi )$ and $-f(\pi )$. Hence the Fourier sine series converges toAt an endpoint this equals $f$ precisely when $f$ is zero there. One can also see the zero endpoint values directly: every partial sum consists of sines that vanish at both endpoints.