![Dirichlet 3](https://upload.wikimedia.org/wikipedia/commons/0/09/Dirichlet_3.jpeg)
In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, there are several
integral
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
s known as the Dirichlet integral, after the German mathematician
Peter Gustav Lejeune Dirichlet
Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and ...
, one of which is the
improper integral
In mathematical analysis, an improper integral is the limit of a definite integral as an endpoint of the interval(s) of integration approaches either a specified real number or positive or negative infinity; or in some instances as both endpoin ...
of the
sinc function
In mathematics, physics and engineering, the sinc function, denoted by , has two forms, normalized and unnormalized..
In mathematics, the historical unnormalized sinc function is defined for by
\operatornamex = \frac.
Alternatively, the u ...
over the positive real line:
:
This integral is not
absolutely convergent
In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is said ...
, meaning
is not Lebesgue-integrable, and so the Dirichlet integral is undefined in the sense of
Lebesgue integration
In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Lebe ...
. It is, however, defined in the sense of the improper
Riemann integral
In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Göt ...
or the generalized Riemann or
Henstock–Kurzweil integral
In mathematics, the Henstock–Kurzweil integral or generalized Riemann integral or gauge integral – also known as the (narrow) Denjoy integral (pronounced ), Luzin integral or Perron integral, but not to be confused with the more general wide ...
. This can be seen by using
Dirichlet's test for improper integrals. Although the
sine integral
In mathematics, trigonometric integrals are a indexed family, family of integrals involving trigonometric functions.
Sine integral
The different sine integral definitions are
\operatorname(x) = \int_0^x\frac\,dt
\operatorname(x) = -\int ...
, an
antiderivative
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolically ...
of the sinc function, is not an
elementary function
In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponen ...
, the value of the integral (in the Riemann or Henstock sense) can be derived using various ways, including the Laplace transform, double integration, differentiating under the integral sign, contour integration, and the Dirichlet kernel.
Evaluation
Laplace transform
Let
be a function defined whenever
. Then its
Laplace transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform
In mathematics, an integral transform maps a function from its original function space into another function space via integra ...
is given by
:
if the integral exists.
A property of the
Laplace transform useful for evaluating improper integrals is
:
provided
exists.
In what follows, one needs the result
, which is the Laplace transform of the function
(see the section 'Differentiating under the integral sign' for a derivation) as well as a version of
Abel's theorem
In mathematics, Abel's theorem for power series relates a limit of a power series to the sum of its coefficients. It is named after Norwegian mathematician Niels Henrik Abel.
Theorem
Let the Taylor series
G (x) = \sum_^\infty a_k x^k
be a powe ...
(a consequence of the
final value theorem for the Laplace transform).
Therefore,
:
Double integration
Evaluating the Dirichlet integral using the Laplace transform is equivalent to calculating the same double definite integral by changing the
order of integration
In statistics, the order of integration, denoted ''I''(''d''), of a time series is a summary statistic, which reports the minimum number of differences required to obtain a covariance-stationary series.
Integration of order ''d''
A time seri ...
, namely,
:
:
Differentiation under the integral sign (Feynman's trick)
First rewrite the integral as a function of the additional variable
, namely, the Laplace transform of
. So let
:
In order to evaluate the Dirichlet integral, we need to determine
. The continuity of
can be justified by applying the
dominated convergence theorem
In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the ''L''1 norm. Its power and utility are two of the primary ...
after integration by parts. Differentiate with respect to
and apply the
Leibniz rule for differentiating under the integral sign to obtain
:
Now, using Euler's formula
one can express the sine function in terms of complex exponentials:
:
Therefore,
:
Integrating with respect to
gives
:
where
is a constant of integration to be determined. Since
using the principal value. This means that for
:
Finally, by continuity at
, we have
, as before.
Complex integration
The same result can be obtained by complex integration. Consider
:
As a function of the complex variable
, it has a simple pole at the origin, which prevents the application of
Jordan's lemma In complex analysis, Jordan's lemma is a result frequently used in conjunction with the residue theorem to evaluate contour integrals and improper integrals. The lemma is named after the French mathematician Camille Jordan.
Statement
Consider a ...
, whose other hypotheses are satisfied.
Define then a new function
:
The pole has been moved away from the real axis, so
can be integrated along the semicircle of radius
centered at
and closed on the real axis. One then takes the limit
.
The complex integral is zero by the residue theorem, as there are no poles inside the integration path
:
The second term vanishes as
goes to infinity. As for the first integral, one can use one version of the
Sokhotski–Plemelj theorem
The Sokhotski–Plemelj theorem (Polish spelling is ''Sochocki'') is a theorem in complex analysis, which helps in evaluating certain integrals. The real-line version of it ( see below) is often used in physics, although rarely referred to by nam ...
for integrals over the real line: for a
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
-valued function defined and continuously differentiable on the real line and real constants
and
with
one finds
:
where
denotes the
Cauchy principal value
In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined.
Formulation
Depending on the type of singularity in the integrand , ...
. Back to the above original calculation, one can write
:
By taking the imaginary part on both sides and noting that the function
is even, we get
:
Finally,
:
Alternatively, choose as the integration contour for
the union of upper half-plane semicircles of radii
and
together with two segments of the real line that connect them. On one hand the contour integral is zero, independently of
and
; on the other hand, as
and
the integral's imaginary part converges to
(here
is any branch of logarithm on upper half-plane), leading to
.
Dirichlet kernel
Let
:
be the
Dirichlet kernel In mathematical analysis, the Dirichlet kernel, named after the German mathematician Peter Gustav Lejeune Dirichlet, is the collection of periodic functions defined as
D_n(x)= \sum_^n e^ = \left(1+2\sum_^n\cos(kx)\right)=\frac,
where is any nonneg ...
.
It immediately follows that
Define
:
Clearly,
is continuous when
, to see its continuity at 0 apply
L'Hopital's Rule:
:
Hence,
fulfills the requirements of the
Riemann-Lebesgue Lemma. This means
:
(The form of the Riemann-Lebesgue Lemma used here is proven in the article cited.)
We would like to say that
:
In order to do so, however, we must justify switching the real limit in
to the integral limit in
. This is in fact justified if we can show the limit does exist, which we do now.
Using
integration by parts
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. ...
, we have:
:
Now, as
and
the term on the left converges with no problem. See the
list of limits of trigonometric functions. We now show that
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,
:
Therefore,
:
Splitting the integral into pieces, we have
:
for some constant
. This shows that the integral is absolutely integrable, which implies the original integral exists, and switching from
to
was in fact justified, and the proof is complete.
See also
*
Dirichlet distribution
In probability and statistics, the Dirichlet distribution (after Peter Gustav Lejeune Dirichlet), often denoted \operatorname(\boldsymbol\alpha), is a family of continuous multivariate probability distributions parameterized by a vector \boldsymb ...
*
Dirichlet principle
In mathematics, and particularly in potential theory, Dirichlet's principle is the assumption that the minimizer of a certain energy functional is a solution to Poisson's equation.
Formal statement
Dirichlet's principle states that, if the functio ...
*
Sinc function
In mathematics, physics and engineering, the sinc function, denoted by , has two forms, normalized and unnormalized..
In mathematics, the historical unnormalized sinc function is defined for by
\operatornamex = \frac.
Alternatively, the u ...
*
Fresnel integral
250px,
Plots of and . The maximum of is about . If the integrands of and were defined using instead of , then the image would be scaled vertically and horizontally (see below).
The Fresnel integrals and are two transcendental functions n ...
References
External links
*
{{Peter Gustav Lejeune Dirichlet
Special functions
Integral calculus
Mathematical physics