sine integral at infinity

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

$${\int }_0^{\mathrm{\infty }}\frac{\sin x}{x}𝑑x=\underset{x\to \mathrm{\infty }}{lim}\mathrm{Si}x,$$

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 $t$:

$$ℒ\{{\int }_0^{\mathrm{\infty }}\frac{1}{x}\sin txdx\}={\int }_0^{\mathrm{\infty }}\frac{1}{x}\cdot \frac{x}{s^2+x^2}𝑑x={\int }_0^{\mathrm{\infty }}\frac{dx}{s^2+x^2}=\frac{1}{s}\underset{x=0}{\overset{\mathrm{\infty }}{/}}\arctan \frac{x}{s}=\frac{\pi }{2}\cdot \frac{1}{s}$$

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  $ℒ\{1\}=\frac{1}{s}$.  Thus we have the result

${\displaystyle {\int }_0^{\mathrm{\infty }}}{\displaystyle \frac{\sin x}{x}}𝑑x={\displaystyle \frac{\pi }{2}}.$ (1)

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

$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\mathrm{sinc}xdx=\pi ;$$

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

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

${\displaystyle {\int }_0^{\mathrm{\infty }}}{\displaystyle \frac{\sin ax}{x}}dx={\displaystyle \frac{\pi }{2}}\mathit{\hspace{1em}\hspace{1em}}(a>0)$ (2)

and to

${\displaystyle {\int }_0^{\mathrm{\infty }}}{\displaystyle \frac{\sin ax}{x}}dx=(\text{sgn}a){\displaystyle \frac{\pi }{2}}\mathit{\hspace{1em}\hspace{1em}}(a\in ℝ).$ (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