= Solution
{c}
<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.<Théorie analytique de la chaleur by Joseph Fourier/Propagation de la chaleur dans un solide rectangle infini>[§233, integral formula; §§163–169, the boundary-value problem motivating the expansions]{ref}
Use a separate integration variable $t$ to unpack the printed notation. The proposed <Fourier sine series> has <coefficients>
$$
\boxed{b_n=\frac{2}{\pi}\int_0^\pi f(t)\sin(nt)\,dt,\qquad
f(x)\sim\sum_{n=1}^{\infty}b_n\sin(nx).}
$$
The 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> relation
$$
\int_0^\pi\sin(nt)\sin(mt)\,dt=
\begin{cases}\pi/2&n=m,\\0&n\ne m.\end{cases}
$$
The <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 therefore
$$
1=\frac4\pi\left(\sin x+\frac{\sin3x}{3}+\frac{\sin5x}{5}+\cdots\right),\qquad0<x<\pi.
$$
Every 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.<Sur la convergence des séries trigonométriques by Peter Gustav Lejeune Dirichlet>[1829, opening discussion and concluding convergence statement]{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> produces
$$
u(x,\tau)=\sum_{n=1}^{\infty}b_n e^{-\kappa n^2\tau}\sin(nx),\qquad\kappa>0.
$$
Each 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. \b[The representation of arbitrary data makes the physical solution possible; the proof of that representation remains a separate task.]
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