Dirichlet integral

Not to be confused withDirichlet energy.

Peter Gustav Lejeune Dirichlet

Inmathematics,there are severalintegralsknown as theDirichlet integral,after the German mathematicianPeter Gustav Lejeune Dirichlet,one of which is theimproper integralof thesinc functionover the positive real line:

$$\int _{0}^{\infty }{\frac {\sin x}{x}}\,dx={\frac {\pi }{2}}.$$

This integral is notabsolutely convergent,meaning$\left|{\frac {\sin x}{x}}\right|$has infinite Lebesgue or Riemann improper integral over the positive real line, so the sinc function is notLebesgue integrableover the positive real line. The sinc function is, however, integrable in the sense of the improperRiemann integralor the generalized Riemann orHenstock–Kurzweil integral.This can be seen by usingDirichlet's test for improper integrals.

It is a good illustration of special techniques for evaluating definite integrals, particularly when it is not useful to directly apply thefundamental theorem of calculusdue to the lack of an elementaryantiderivativefor the integrand, as thesine integral,an antiderivative of the sinc function, is not anelementary function.In this case, the improper definite integral can be determined in several ways: the Laplace transform, double integration, differentiating under the integral sign, contour integration, and the Dirichlet kernel.

Evaluation

Laplace transform

Let$f(t)$be a function defined whenever$t\geq 0.$Then itsLaplace transformis given by $${\mathcal {L}}\{f(t)\}=F(s)=\int _{0}^{\infty }e^{-st}f(t)\,dt,$$ if the integral exists.

A property of theLaplace transform useful for evaluating improper integralsis $${\mathcal {L}}\left[{\frac {f(t)}{t}}\right]=\int _{s}^{\infty }F(u)\,du,$$ provided$\lim _{t\to 0}{\frac {f(t)}{t}}$exists.

In what follows, one needs the result${\mathcal {L}}\{\sin t\}={\frac {1}{s^{2}+1}},$which is the Laplace transform of the function$\sin t$(see the section 'Differentiating under the integral sign' for a derivation) as well as a version ofAbel's theorem(a consequence of thefinal value theorem for the Laplace transform).

Therefore, $${\begin{aligned}\int _{0}^{\infty }{\frac {\sin t}{t}}\,dt&=\lim _{s\to 0}\int _{0}^{\infty }e^{-st}{\frac {\sin t}{t}}\,dt=\lim _{s\to 0}{\mathcal {L}}\left[{\frac {\sin t}{t}}\right]\\[6pt]&=\lim _{s\to 0}\int _{s}^{\infty }{\frac {du}{u^{2}+1}}=\lim _{s\to 0}\arctan u{\Biggr |}_{s}^{\infty }\\[6pt]&=\lim _{s\to 0}\left[{\frac {\pi }{2}}-\arctan(s)\right]={\frac {\pi }{2}}.\end{aligned}}$$

Double integration

Evaluating the Dirichlet integral using the Laplace transform is equivalent to calculating the same double definite integral by changing theorder of integration,namely, $$\left(I_{1}=\int _{0}^{\infty }\int _{0}^{\infty }e^{-st}\sin t\,dt\,ds\right)=\left(I_{2}=\int _{0}^{\infty }\int _{0}^{\infty }e^{-st}\sin t\,ds\,dt\right),$$ $$\left(I_{1}=\int _{0}^{\infty }{\frac {1}{s^{2}+1}}\,ds={\frac {\pi }{2}}\right)=\left(I_{2}=\int _{0}^{\infty }{\frac {\sin t}{t}}\,dt\right),{\text{ provided }}s>0.$$ The change of order is justified by the fact that for all$s>0$,the integral is absolutely convergent.

Differentiation under the integral sign (Feynman's trick)

First rewrite the integral as a function of the additional variable$s,$namely, the Laplace transform of${\frac {\sin t}{t}}.$So let $$f(s)=\int _{0}^{\infty }e^{-st}{\frac {\sin t}{t}}\,dt.$$

In order to evaluate the Dirichlet integral, we need to determine$f(0).$The continuity of$f$can be justified by applying thedominated convergence theoremafter integration by parts. Differentiate with respect to$s>0$and apply theLeibniz rule for differentiating under the integral signto obtain $${\begin{aligned}{\frac {df}{ds}}&={\frac {d}{ds}}\int _{0}^{\infty }e^{-st}{\frac {\sin t}{t}}\,dt=\int _{0}^{\infty }{\frac {\partial }{\partial s}}e^{-st}{\frac {\sin t}{t}}\,dt\\[6pt]&=-\int _{0}^{\infty }e^{-st}\sin t\,dt.\end{aligned}}$$

Now, using Euler's formula$e^{it}=\cos t+i\sin t,$one can express the sine function in terms of complex exponentials: $$\sin t={\frac {1}{2i}}\left(e^{it}-e^{-it}\right).$$

Therefore, $${\begin{aligned}{\frac {df}{ds}}&=-\int _{0}^{\infty }e^{-st}\sin t\,dt=-\int _{0}^{\infty }e^{-st}{\frac {e^{it}-e^{-it}}{2i}}dt\\[6pt]&=-{\frac {1}{2i}}\int _{0}^{\infty }\left[e^{-t(s-i)}-e^{-t(s+i)}\right]dt\\[6pt]&=-{\frac {1}{2i}}\left[{\frac {-1}{s-i}}e^{-t(s-i)}-{\frac {-1}{s+i}}e^{-t(s+i)}\right]_{0}^{\infty }\\[6pt]&=-{\frac {1}{2i}}\left[0-\left({\frac {-1}{s-i}}+{\frac {1}{s+i}}\right)\right]=-{\frac {1}{2i}}\left({\frac {1}{s-i}}-{\frac {1}{s+i}}\right)\\[6pt]&=-{\frac {1}{2i}}\left({\frac {s+i-(s-i)}{s^{2}+1}}\right)=-{\frac {1}{s^{2}+1}}.\end{aligned}}$$

Integrating with respect to$s$gives $$f(s)=\int {\frac {-ds}{s^{2}+1}}=A-\arctan s,$$

where$A$is a constant of integration to be determined. Since$\lim _{s\to \infty }f(s)=0,$$A=\lim _{s\to \infty }\arctan s={\frac {\pi }{2}},$using the principal value. This means that for$s>0$ $$f(s)={\frac {\pi }{2}}-\arctan s.$$

Finally, by continuity at$s=0,$we have$f(0)={\frac {\pi }{2}}-\arctan(0)={\frac {\pi }{2}},$as before.

Complex contour integration

Consider$$f(z)={\frac {e^{iz}}{z}}.$$

As a function of the complex variable$z,$it has a simple pole at the origin, which prevents the application ofJordan's lemma,whose other hypotheses are satisfied.

Define then a new function $$g(z)={\frac {e^{iz}}{z+i\varepsilon }}.$$

The pole has been moved to the negative imaginary axis, so$g(z)$can be integrated along the semicircle$\gamma$of radius$R$centered at$z=0$extending in the positive imaginary direction, and closed along the real axis. One then takes the limit$\varepsilon \to 0.$

The complex integral is zero by theresidue theorem,as there are no poles inside the integration path$\gamma$: $$0=\int _{\gamma }g(z)\,dz=\int _{-R}^{R}{\frac {e^{ix}}{x+i\varepsilon }}\,dx+\int _{0}^{\pi }{\frac {e^{i(Re^{i\theta }+\theta )}}{Re^{i\theta }+i\varepsilon }}iR\,d\theta.$$

The second term vanishes as$R$goes to infinity. As for the first integral, one can use one version of theSokhotski–Plemelj theoremfor integrals over the real line: for acomplex-valued functionfdefined and continuously differentiable on the real line and real constants$a$and$b$with$a<0<b$one finds $$\lim _{\varepsilon \to 0^{+}}\int _{a}^{b}{\frac {f(x)}{x\pm i\varepsilon }}\,dx=\mp i\pi f(0)+{\mathcal {P}}\int _{a}^{b}{\frac {f(x)}{x}}\,dx,$$

where${\mathcal {P}}$denotes theCauchy principal value.Back to the above original calculation, one can write $$0={\mathcal {P}}\int {\frac {e^{ix}}{x}}\,dx-\pi i.$$

By taking the imaginary part on both sides and noting that the function$\sin(x)/x$is even, we get $$\int _{-\infty }^{+\infty }{\frac {\sin(x)}{x}}\,dx=2\int _{0}^{+\infty }{\frac {\sin(x)}{x}}\,dx.$$

Finally, $$\lim _{\varepsilon \to 0}\int _{\varepsilon }^{\infty }{\frac {\sin(x)}{x}}\,dx=\int _{0}^{\infty }{\frac {\sin(x)}{x}}\,dx={\frac {\pi }{2}}.$$

Alternatively, choose as the integration contour for$f$the union of upper half-plane semicircles of radii$\varepsilon$and$R$together with two segments of the real line that connect them. On one hand the contour integral is zero, independently of$\varepsilon$and$R;$on the other hand, as$\varepsilon \to 0$and$R\to \infty$the integral's imaginary part converges to$2I+\Im {\big (}\ln 0-\ln(\pi i){\big )}=2I-\pi$(here$\ln z$is any branch of logarithm on upper half-plane), leading to$I={\frac {\pi }{2}}.$

Dirichlet kernel

Consider the well-known formula for theDirichlet kernel:$$D_{n}(x)=1+2\sum _{k=1}^{n}\cos(2kx)={\frac {\sin[(2n+1)x]}{\sin(x)}}.$$

It immediately follows that:$$\int _{0}^{\frac {\pi }{2}}D_{n}(x)\,dx={\frac {\pi }{2}}.$$

Define $$f(x)={\begin{cases}{\frac {1}{x}}-{\frac {1}{\sin(x)}}&x\neq 0\\[6pt]0&x=0\end{cases}}$$

Clearly,$f$is continuous when$x\in (0,\pi /2];$to see its continuity at 0 applyL'Hopital's Rule: $$\lim _{x\to 0}{\frac {\sin(x)-x}{x\sin(x)}}=\lim _{x\to 0}{\frac {\cos(x)-1}{\sin(x)+x\cos(x)}}=\lim _{x\to 0}{\frac {-\sin(x)}{2\cos(x)-x\sin(x)}}=0.$$

Hence,$f$fulfills the requirements of theRiemann-Lebesgue Lemma.This means: $$\lim _{\lambda \to \infty }\int _{0}^{\pi /2}f(x)\sin(\lambda x)dx=0\quad \Longrightarrow \quad \lim _{\lambda \to \infty }\int _{0}^{\pi /2}{\frac {\sin(\lambda x)}{x}}dx=\lim _{\lambda \to \infty }\int _{0}^{\pi /2}{\frac {\sin(\lambda x)}{\sin(x)}}dx.$$

(The form of the Riemann-Lebesgue Lemma used here is proven in the article cited.)

We would like to compute: $${\begin{aligned}\int _{0}^{\infty }{\frac {\sin(t)}{t}}dt=&\lim _{\lambda \to \infty }\int _{0}^{\lambda {\frac {\pi }{2}}}{\frac {\sin(t)}{t}}dt\\[6pt]=&\lim _{\lambda \to \infty }\int _{0}^{\frac {\pi }{2}}{\frac {\sin(\lambda x)}{x}}dx\\[6pt]=&\lim _{\lambda \to \infty }\int _{0}^{\frac {\pi }{2}}{\frac {\sin(\lambda x)}{\sin(x)}}dx\\[6pt]=&\lim _{n\to \infty }\int _{0}^{\frac {\pi }{2}}{\frac {\sin((2n+1)x)}{\sin(x)}}dx\\[6pt]=&\lim _{n\to \infty }\int _{0}^{\frac {\pi }{2}}D_{n}(x)dx={\frac {\pi }{2}}\end{aligned}}$$

However, we must justify switching the real limit in$\lambda$to the integral limit in$n,$which will follow from showing that the limit does exist.

Usingintegration by parts,we have: $$\int _{a}^{b}{\frac {\sin(x)}{x}}dx=\int _{a}^{b}{\frac {d(1-\cos(x))}{x}}dx=\left.{\frac {1-\cos(x)}{x}}\right|_{a}^{b}+\int _{a}^{b}{\frac {1-\cos(x)}{x^{2}}}dx$$

Now, as$a\to 0$and$b\to \infty$the term on the left converges with no problem. See thelist of limits of trigonometric functions.We now show that$\int _{-\infty }^{\infty }{\frac {1-\cos(x)}{x^{2}}}dx$is absolutely integrable, which implies that the limit exists.

First, we seek to bound the integral near the origin. Using the Taylor-series expansion of the cosine about zero, $$1-\cos(x)=1-\sum _{k\geq 0}{\frac {{(-1)^{(k+1)}}x^{2k}}{2k!}}=\sum _{k\geq 1}{\frac {{(-1)^{(k+1)}}x^{2k}}{2k!}}.$$

Therefore, $$\left|{\frac {1-\cos(x)}{x^{2}}}\right|=\left|-\sum _{k\geq 0}{\frac {x^{2k}}{2(k+1)!}}\right|\leq \sum _{k\geq 0}{\frac {|x|^{k}}{k!}}=e^{|x|}.$$

Splitting the integral into pieces, we have $$\int _{-\infty }^{\infty }\left|{\frac {1-\cos(x)}{x^{2}}}\right|dx\leq \int _{-\infty }^{-\varepsilon }{\frac {2}{x^{2}}}dx+\int _{-\varepsilon }^{\varepsilon }e^{|x|}dx+\int _{\varepsilon }^{\infty }{\frac {2}{x^{2}}}dx\leq K,$$

for some constant$K>0.$This shows that the integral is absolutely integrable, which implies the original integral exists, and switching from$\lambda$to$n$was in fact justified, and the proof is complete.

See also

• Dirichlet distribution
• Dirichlet principle
• Sinc function
• Fresnel integral

External links

• Weisstein, Eric W."Dirichlet Integrals".MathWorld.