Dirichlet discontinuous multiplier

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The integral

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

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.


[1] P.G.L. Dirichlet, "Werke" , 1 , Chelsea, reprint (1969)
[2] G.M. Fichtenholz, "Differential und Integralrechnung" , 2–3 , Deutsch. Verlag Wissenschaft. (1964)
How to Cite This Entry:
Dirichlet discontinuous multiplier. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by T.P. Lukashenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article