Dirichlet discontinuous multiplier

The integral

$$\int\limits_0^\infty\frac{\sin\alpha x}{x}\cos\beta xdx=\begin{cases}\frac\pi2&\text{if }\beta<\alpha,\\\frac\pi4&\text{if }\beta=\alpha,\\0&\text{if }\beta>\alpha,\end{cases}$$

which is a discontinuous function of the parameters $\alpha>0$ and $\beta>0$. Used by P.G.L. Dirichlet in his studies on the attraction of ellipsoids [1]. The integral was encountered earlier in the work of J. Fourier, S. Poisson and A. Legendre.

References

[1]	P.G.L. Dirichlet, "Werke" , 1 , Chelsea, reprint (1969)
[2]	G.M. Fichtenholz, "Differential und Integralrechnung" , 2–3 , Deutsch. Verlag Wissenschaft. (1964)