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In [[mathematics]], there are several [[integral]]s known as the '''''Dirichlet integral''''',  after the German mathematician [[Peter Gustav Lejeune Dirichlet]].


One of those is
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: <math>\int_0^\infty \frac{\sin \omega}{\omega}\,d\omega = \frac{\pi}{2}</math>
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This integral is not [[absolutely convergent]], and so the integral is ''not even defined'' in the sense of [[Lebesgue integration]], but it ''is'' defined in the sense of the [[Riemann integral]] or the [[Henstock–Kurzweil integral]].<ref>Robert G. Bartle, [http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=2900 Return to the Riemann Integral], ''The American Mathematical Monthly'', vol. 103, 1996, pp. 625-632.</ref> The value of the integral (in the Riemann or Henstock sense) can be derived in various ways. For example, the value can be determined from attempts to evaluate a double [[improper integral]], or by using [[differentiation under the integral sign]].
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=== Double improper integral method ===
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Pre-knowledge of properties of [[Laplace transform#Evaluating improper integrals|Laplace transforms]] allows us to evaluate this Dirichlet integral succinctly in the following manner:
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: <math>\int_0^\infty\frac{\sin t}{t}\, dt=\int_{0}^{\infty}\mathcal{L}\{\sin t\}(s)\; ds=\int_{0}^{\infty}\frac{1}{s^{2}+1}\, ds=\arctan s\bigg|_{0}^{\infty}=\frac{\pi}{2},
</math>
where <math>\mathcal{L}\{\sin t\}(s)</math> is the Laplace transform of the function, <math>\sin t</math> that returns the function <math>\tfrac{1}{s^{2}+1}</math>, a function of the Laplace transform variable, ''s''. This is equivalent to attempting to evaluate the same double definite integral in two different ways, by reversal of the [[order of integration (calculus)|order of integration]], ''viz.'',
: <math>\left ( I_1=\int_0^\infty {\int _0^\infty e^{-st} \sin t\, dt}\, ds\right ) = \left ( I_2=\int_0^\infty {\int _0^\infty e^{-st} \sin t\, ds} \, dt = \int_0^\infty \sin t{\int _0^\infty e^{-st}\, ds} \, dt\right ),</math>
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=== Differentiation under the integral sign ===
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First rewrite the integral as a function of variable <math>\!a</math>. Let
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: <math>f(a)=\int_0^\infty e^{-a\omega} \frac{\sin \omega}{\omega} d\omega ;</math>


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Differentiate with respect to <math>\!a</math> and apply the [[Leibniz integral rule]] to obtain:
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: <math>\frac{df}{da}=\frac{d}{da}\int_0^\infty e^{-a\omega} \frac{\sin \omega}{\omega} d\omega = \int_0^\infty  \frac{\partial}{\partial a}e^{-a\omega}\frac{\sin \omega}{\omega} d\omega = -\int_0^\infty e^{-a\omega} \sin \omega \,d\omega = -\mathcal{L}\{\sin \omega\}(a).</math>


This integral was evaluated without proof, above, based on Laplace transform tables; we derive it this time. It is made much simpler by recalling Euler's formula,
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: <math>\! e^{i\omega}=\cos \omega + i\sin \omega ,</math>
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Integrating with respect to <math>\!a</math>:
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where <math>\! A</math> is a constant to be determined. As,
 
: <math>f(+\infty)=0 \therefore A = \arctan (+\infty) = \frac{\pi}{2} + m\pi,</math>
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: <math>\therefore f(0)=\lim _{a \to 0^+} f(a) = \frac{\pi}{2} + m\pi - \arctan 0 = \frac{\pi}{2} + n\pi,</math>
 
for some integers ''m'' & ''n''. It is easy to show that <math>\!n</math> has to be zero, by analyzing easily observed bounds for this integral:
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: <math>0<\int _0^\infty \frac {\sin x}{x}dx < \int _0^\pi \frac {\sin x}{x}dx < \pi</math>
 
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The left and right bounds can be derived by dividing the integrated region [0:∞] to periodic intervals, over which the integrals have zero value. Left bound, ∫[0:∞]sinx/x dx = ∫[0:2π]sinx/x dx + ∫[2π:4π]sinx/x dx + ...  > ∫[0:2π]sinx/(2π) dx +  ∫[2π:4π]sinx/(4π) dx + ...  = 0. Right bound, ∫[0:∞]sinx/x dx = ∫[0:π] sinx/x dx + ∫[π:∞] sinx/x dx < ∫[0:π] sinx/x dx + ∫[π:∞] sinx dx. The second term is zero, which can be proved using the same approach as for the left bound. The first term, ∫[0:π] sinx/x dx < ∫[0:π] dx = π.
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Extending this result further, with the introduction of another variable, first noting that <math>\! {\sin x}/{x}</math> is an even function and therefore
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then:
: <math>\int_0^\infty \frac{\sin b\,\omega}{\omega}\,d\omega = \int_0^{b\,\infty} \frac{\sin b\,\omega}{b\,\omega}\,d(b\,\omega) = \int_0^{\sgn b\times\infty} \frac{\sin x}{x}\,dx = \sgn b \int_0^\infty \frac{\sin x}{x}\,dx = \frac{\pi}{2}\,\sgn b</math>


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The same result can be obtained via complex integration. Let's consider
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As a function of the complex variable z, it has a simple pole at the origin, which prevents the application of [[Jordan's lemma]], whose other hypotheses are satisfied. We shall then define a new function<ref>Appel, Walter. ''Mathematics for Physics and Physicists''. Princeton University Press, 2007, p. 226.</ref> g(z) as follows
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: <math> 0 = \int_\gamma g(z) dz = \int_{-R}^R \frac{e^{ix}}{x +i\epsilon} dx + \int_{0}^{\pi} \frac{e^{i(Re^{i\theta} + \theta)}}{Re^{i\theta} +i\epsilon} iR d\theta </math>
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The second term vanishes as R goes to infinity; for arbitrarily small <math> \epsilon </math>, the [[Sokhotski–Plemelj theorem]] applied to the first one yields
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: <math> \lim_{\epsilon\rightarrow 0}\int_\epsilon^{\infty} \frac{\sin(x)}{x} dx = \int_0^{\infty} \frac{\sin(x)}{x} dx = \frac{\pi}{2} </math>
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== See also ==
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* [[Dirichlet principle]]
  <li>[http://aot.jp/bbpress/profile.php?id=551135 http://aot.jp/bbpress/profile.php?id=551135]</li>
 
 
==Notes==
  <li>[http://www.southernfootballhistory.com/phrum/read.php?49,148554 http://www.southernfootballhistory.com/phrum/read.php?49,148554]</li>
 
 
{{Reflist}}
</ul>
 
== External links ==
 
* {{MathWorld | urlname=DirichletIntegrals | title=Dirichlet Integrals}}
 
[[Category:Calculus]]
[[Category:Special functions]]
[[Category:Integral calculus]]

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