Dirichlet integral's history

Question

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?

Answer 1

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.