# sine integral at infinity

The value of the improper integral (one of the Dirichlet integrals)

 $\int_{0}^{\infty}\frac{\sin{x}}{x}\,dx=\lim_{x\to\infty}\operatorname{Si}{x},$

where Si means the sine integral (http://planetmath.org/SineIntegral) function, is most simply determined by using Laplace transform which may be aimed to the integrand (see integration of Laplace transform with respect to parameter).  Therefore the integrand must be equipped with an additional parametre $t$:

 $\mathcal{L}\{\int_{0}^{\infty}\frac{1}{x}\sin{tx}\,dx\}=\int_{0}^{\infty}\frac% {1}{x}\!\cdot\!\frac{x}{s^{2}+x^{2}}\,dx=\int_{0}^{\infty}\!\frac{dx}{s^{2}+x^% {2}}=\frac{1}{s}\!\operatornamewithlimits{\Big{/}}_{\!\!\!x=0}^{\,\quad\infty}% \!\arctan{\frac{x}{s}}=\frac{\pi}{2}\!\cdot\!\frac{1}{s}$

The obtained transform $\frac{\pi}{2}\!\cdot\!\frac{1}{s}$ corresponds (see the inverse Laplace transformation) to the function  $t\mapsto\frac{\pi}{2}$  because  $\mathcal{L}\{1\}=\frac{1}{s}$.  Thus we have the result

 $\displaystyle\int_{0}^{\infty}\frac{\sin{x}}{x}\,dx\;=\;\frac{\pi}{2}.$ (1)

Note 1.  Since  $x\mapsto\frac{\sin{x}}{x}$  or  $x\mapsto\operatorname{sinc}{x}$  is an even function, the result (1) may be written also

 $\int_{-\infty}^{\infty}\operatorname{sinc}{x}\,dx=\pi;$

see the $\operatorname{sinc}$-function (http://planetmath.org/SincFunction).

Note 2.  The result (1) may be easily generalised to

 $\displaystyle\int_{0}^{\infty}\frac{\sin{ax}}{x}\,dx\;=\;\frac{\pi}{2}\qquad(a% >0)$ (2)

and to

 $\displaystyle\int_{0}^{\infty}\frac{\sin{ax}}{x}\,dx\;=\;(\mbox{sgn}\,a)\frac{% \pi}{2}\qquad(a\in\mathbb{R}).$ (3)
 Title sine integral at infinity Canonical name SineIntegralAtInfinity Date of creation 2013-03-22 15:17:22 Last modified on 2013-03-22 15:17:22 Owner pahio (2872) Last modified by pahio (2872) Numerical id 18 Author pahio (2872) Entry type Derivation Classification msc 44A10 Classification msc 30A99 Synonym limit of sine integral Related topic SineIntegral Related topic SincFunction Related topic SubstitutionNotation Related topic IncompleteGammaFunction Related topic ExampleOfSummationByParts Related topic SignumFunction