PlanetPhysics/Laplace Transform of Diracs Delta

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A Dirac  symbol can be interpreted as a linear functional, i.e. a linear mapping from a function space, consisting e.g. of certain real functions, to  (or ), having the property

One may think this as the inner product of a function and another "function" , when the well-known formula is true.\, Applying this to\, ,\, one gets

i.e. the Laplace transform

By the delay theorem, this result may be generalised to \\

When introducing a so-called "Dirac delta function", for example

as an "approximation" of Dirac delta, we obtain the Laplace transform As the Taylor expansion shows, we then have according to ref.(2).

Laplace transform of Dirac deltaEdit

The Dirac delta , , can be correctly defined as a linear functional, i.e. a linear mapping from a function space, consisting e.g. of certain real functions, to (or ), having the property One may think of this as an inner product of a function and another "function" , when the well-known formula holds.\, By applying this to \, ,\, one gets i.e. the Laplace transform

By the delay theorem, this result may be generalised to:

All SourcesEdit

[1][2][3]

ReferencesEdit

  1. Schwartz, L. (1950--1951), Th\'eorie des distributions, vols. 1--2, Hermann: Paris.
  2. W. Rudin, Functional Analysis , McGraw-Hill Book Company, 1973.
  3. L. H\"ormander, {\em The Analysis of Linear Partial Differential Operators I, (Distribution theory and Fourier Analysis)}, 2nd ed, Springer-Verlag, 1990.