Dirac delta function

The delta function (also δ - function; Dirac function, pulse, pulse, shock ( by Paul Dirac ), shock function, needle pulse, pulse function or unit impulse function called ) as a mathematical concept is a special irregular distribution with compact support. She has in the mathematics and physics of fundamental importance. Their usual formula symbol δ ( small delta ).

  • 5.1 irregularity
  • 5.2 derivations 5.2.1 Derivation of the delta function
  • 5.2.2 Derivation of the Dirac sequence
  • 5.2.3 Derivation of Heaviside distribution
  • 5.3.1 Fourier transform
  • 5.3.2 Laplace transform
  • 5.3.3 Note about the representation
  • 5.3.4 Transformation of the shifted delta function
  • 7.1 Delta function in curvilinear coordinate systems 7.1.1 Examples

Definition

The delta distribution is a continuous linear mapping from a function space of test functions in the underlying field:

The test function space for the delta function is the space of infinitely differentiable functions with or open. Thus corresponds to either the real or the complex numbers.

The delta distribution assigns each infinitely differentiable function is a real or complex number, namely the evaluation of the function at the point 0, the value that the delta function provides the application on a test function to write ( with the notation of the dual pairing) as

Or as

This notation is actually not correct and only to be understood symbolically, because the delta function is an irregular distribution, ie they can not be represented by a locally integrable function in the above manner. So there is no function that satisfies the above definition ( see below for proof " irregularity "). Especially with technically -oriented applications of the concept are still not mathematically precise terms such as " delta function ", " Dirac function " or " pulse function" in use. When using the integral notation is important to note that it is not a Riemann integral or the Lebesgue integral with respect to the Lebesgue measure, but the evaluation of the functional on the site, so is.

Definition via the Dirac measure

The functional generated by a positive Radon measure ( for ) is a distribution. The delta function is of the following Radon measure - one talks specifically from Diracmaß - created:

A measure can be interpreted physically, such as a mass distribution, or load distribution in the room. Then the delta function corresponds to a point mass of mass 1 or a point charge of charge 1 at the origin.

Located at the positions of point charges, the sum of all shipments remains finite, then ( go through all with ) for a measure on the - algebra of all subsets of defined corresponding to the charge distribution:

Then the corresponding distribution for this measure:

Approximation of the delta function

One can represent the delta function like all other distributions as the limit of a sequence of functions. The amount of the Dirac sequences is the most important class of sequences of functions with which the delta function can be represented. However, there are other consequences that converge to the delta distribution.

Dirac sequence

A sequence of integrable functions is called Dirac sequence if

Applies. Sometimes we mean by a Dirac sequence and only one of the defined here Dirac sequence a special case. If one chooses namely a function with for all and, seating for, then this family of functions satisfies the properties 1 and 2 instead of considering the limit as well as property 3 is satisfied. Therefore, also called the family of functions Dirac sequence.

Comments

The function can now be with a regular distribution

Identify. Only in the limit we obtain the unusual behavior of the delta function

It being noted that the Limes formation is not under the integral, but before that. If one were to draw the limit under the integral, it would be almost zero everywhere, just not at. However, a single dot has the Lebesgue measure zero and the whole integral would vanish.

Clearly, one imagines the delta function as an arbitrarily high and arbitrarily small function which encloses a region with size 1 unit area above the x -axis. You can now function are always narrower and for getting higher - the area must remain constant under 1. There are also multi-dimensional Dirac distributions, these are clearly multidimensional " clubs " with the first volume

Examples of Dirac Follow

The following approximations are given ( Dirac episodes), initially continuously differentiable:

  • Bell functions ( normal distributions )
  • Lorentz curves
  • Fresnel representation

However, it is also possible approximations that are only piecewise continuously differentiable:

  • Rectangular function
  • Triangular function
  • Exponential falling to origin

Other examples

  • The functional consequence of the sinc functions is not a Dirac sequence, as their followers also assume negative values ​​. However, looking at the expression converges for all these result in the distributional sense against the Delta function.

Definition in the nonstandardanalysis

In the nonstandardanalysis can the " delta function " explicitly define a function with the desired properties.

Properties

  • Called convolution property, also shifting property, Siebeigenschaft: The defining property of the delta function
  • Linearity:
  • Translation:
  • Scaling:
  • Dimension
  • Sequential execution:
  • Therefore follows as a special case, the calculation rule

Irregularity

The irregularity ( = singularity ) of the delta function can be demonstrated with a proof by contradiction:

Would regularly adopted, there would be a locally integrable function, ie a function that can be integrated over each interval with respect to the compact Lebesgue measure

Such that for all test functions applies:

In particular, this must apply to the following test function with compact support:

The effect of the delta function to this is:

With the assumed normal distribution

Can perform the following assessment:

Because the integral for ( being a dependent of the function of critical value) of less than 1 ( and converges to 0 for 0 against ). Is obtained, ie a contradiction; Thus, the delta function can not be represented by a locally integrable function. The contradiction arises because the set { 0} is negligible for the Lebesgue measure, but not for the Dirac measure.

Derivations

Derivative of the delta function

The delta function can be differentiated as often as every distribution distributive:

And the th distributive derivation:

The Dirac sequence derivation

The derivatives of the regular distributions can be calculated by means of partial integration ( here exemplarily for the first derivative, analogue for higher )

And result in the limit behavior of distributive derivation:

Derivative of the Heaviside distribution

The Heaviside function is not continuously differentiable, but the distributive derivative exists, this is in fact the delta function:

Since the Heaviside distribution does not have compact support, where the test functions are infinitely differentiable functions with compact support must be, that is, must vanish at infinity

Fourier Laplace transform

Since the delta function has compact support, it is possible to form the Fourier transform of this Laplace. For this

Fourier transformation

Fourier Laplace transform is a special case of the Fourier transform, and thus is also

There are also a factor with the convention of the Fourier transform to be multiplied. In which case, too, the result of the Fourier transform of the delta function. This clearly means the result of the transformation that in the delta distribution, all frequencies are included, with equal strength. The display (in case of the other convention for the prefactor ) is an important in the presentation of the physics delta distribution.

Laplace transform

The Laplace transform of the delta function is obtained as a special case of the Fourier transformation, Laplace. Namely, we also

In contrast to the Fourier transform, there is no other conventions here.

Note about the representation

Often the Fourier or Laplace transform is represented by the normal integral notation. However, these representations are

For the Fourier transformation, or

Only symbolic of the Laplace transform to understand and mathematically undefined.

Transformation of the shifted delta function

It is also possible, the Fourier transform or the Laplace transform to calculate for the shifted delta distribution. It is

Practical Application

Practical significance of the Dirac impulse in the determination of the impulse response in acoustics ( in other branches of physics one speaks also of a size, if it is felt that the tire size of a narrowest - possible distribution is sufficient). So each room has its own sound behavior. With a Dirac pulse ( approximated by a clapping ), this behavior can be determined ( by measuring the " echo ", so the system response ).

Typical, technically feasible Dirac values:

  • Laser pulse technique about 10-100 fs half-width

An important application of the delta function is the solution of inhomogeneous linear differential equations using the method of Green's function.

Multidimensional delta function

In Multidimensional is the space of test functions equal, the space of infinitely often differentiable continuous partial functions.

The delta function has the test function, the following effect:

In the integral notation using translation and scaling:

The " multi-dimensional " delta function can be written as a product of " one-dimensional " delta distributions:

Especially in three dimensions there is a representation of the delta function, which is often used in electrodynamics to represent point charges:

Delta function in curvilinear coordinate systems

In curvilinear coordinate systems, the Jacobian must

Be considered.

The approach

With and leads to the equation

It can be seen, that must apply

In curvilinear coordinate system, the delta distribution must therefore be provided with a pre-factor, corresponding to the inverse of the Funktionaldeterminate.

Examples

In spherical coordinates with and is:

In cylindrical coordinates with and is:

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