Laplace transform of Dirac's delta distribution
LaplaceTransformOfDiracsDelta

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

(0.1)

By the delay theorem, this result may be generalised to


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

   for   for    

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 delta

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

(0.2)

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

Bibliography

1
Schwartz, L. (1950-1951), Théorie des distributions, vols. 1-2, Hermann: Paris.

2
W. Rudin, Functional Analysis, McGraw-Hill Book Company, 1973.

3
L. Hörmander, The Analysis of Linear Partial Differential Operators I, (Distribution theory and Fourier Analysis), 2nd ed, Springer-Verlag, 1990.



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