# Dirichlet integral's history

I was calculating the so-called Dirichlet Integral,

$$\displaystyle \int_{0}^{\infty}\frac{\sin x}{x} \text{d} x$$

and then I wondered about his name and history:

1. Why it has the name of Dirichlet?

2. When was it first calculated?

3. In what context it showed up for the first time and from which interests it needed (if so) to be evaluated?

I don't know any resources (except for Stack Exchange) for researching this questions, and I haven't found anything related to them after surfing along the first 4 pages of results of "dirichlet integral history" in Google.

Does anybody know anything about its history or where to find information about it?

## 1 Answer

It arose in Dirichlet’s famous proof of the convergence of Fourier series (1829, p. 161),1 then again in his “discontinuous factor” method to compute integrals (1839; 1904, pp. 193-195, 353-385)2 and in his “jump function” proof of the central limit theorem (1846; 2011, pp. 69-74).3

Apparently first evaluated by Euler in De valoribus integralium a termino variabilis $x = 0$ usque ad $x = \infty$ extensorum (1781; 1794, pp. 337–345; 1932, pp. lxiii, 217–227);4 Dirichlet’s uses are also foreshadowed in Fourier (1822, pp. 510-512),5 Poisson (1824, p. 276), Bessel (1838, p. 393).

Incidentally, I suspect that calling it the Dirichlet integral is a misnomer introduced by Wikipedia: so far as I can tell, none of the below-quoted authors do that. Rather, they call any integral of the form $\smash{\int_0^\infty f(x)\frac{\sin kx}{x}dx}$ (including of course this one) a Dirichlet integral.

References in English: 1Dirichlet p. 255, Carslaw pp. 200-212, Bottazzini pp. 190-196, Roy pp. 417-421, Petrovic Prob. 9.2.14. 2Price pp. 399-410. 3Fischer pp. 44-52. 4Euler p. 7. 5Fourier §423.

• An entire paper actually exists: Fischer, Hans, The history of the integral $\int_0^\infty \frac{\sin x}{x}\,dx$: a history of analysis in a nutshell, Math. Semesterber. 54, No. 1, 13-30 (2007). ZBL1118.01014. Dec 23, 2017 at 4:59
• I don't speak french nor german, I can't understand the sources. What was being proved exactly in that "Dirichlet’s famous proof of the convergence of Fourier series"? Dec 24, 2017 at 10:02
• @ElíasGuisado I have tried to add references to English expositions / translations. Dec 27, 2017 at 11:35