# Dirichlet's Proof of the Convergence of Fourier Series

Where can I find Dirichlet's proof of the convergence of Fourier series?

• Are you asking for the original source or just some exposition of what textbooks call "Dirichlet's theorem"? Aug 22 '18 at 21:55
• @Conifold Both. Aug 23 '18 at 12:22
• Possible duplicate of Dirichlet integral's history Jul 14 '19 at 2:19

"However, it is a mistake to suppose that Fourier did not establish in a rigorous and conclusive manner that a quite arbitrary function (meaning by this any function capable of being represented by an arc of a continuous curve or by successive portions of diﬀerent continuous curves), could be represented by the series with his name, and it is equally wrong to attribute the ﬁrst rigorous demonstration of this theorem to Dirichlet, whose earliest memoir was published in 1829. A closer examination of Fourier’s work will show that the importance of his investigations merits the fullest recognition, and Darboux, in the latest complete edition of Fourier’s mathematical works points out that the method he employed in the ﬁnal discussion of the general case is perfectly sound and practically identical with that used later by Dirichlet in his classical memoir. In this discussion Fourier followed the line of argument which is now customary in dealing with inﬁnite series. He proved that when the values $$a_n,b_n$$ are inserted in the terms of the series" $$a0 + (a1 \cos x + b1 \sin x) + (a2 \cos 2x + b2 \sin 2x) + …,$$ the sum of the terms up to $$\cos(nx)$$ and $$\sin(nx)$$ is $$\frac{1}{\pi}\int_{-\pi}^{\pi} f(x')\frac{ \sin \frac{1}{2}(2n + 1)(x'−x) }{\sin \frac{1}{2}(x'−x)} dx'$$ He then discussed the limiting value of this sum as n becomes inﬁnite, and thus obtained the sum of the series now called Fourier’s Series."