Laplace–Carson transform

Short description: Variant of the Laplace integral transform

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In mathematics, the Laplace–Carson transform, named for Pierre Simon Laplace and John Renshaw Carson, is an integral transform closely related to the standard Laplace transform. It is defined by multiplying the Laplace transform of a function by the complex variable p.[1] This modification can simplify the analysis of certain functions, particularly the unit step function and Dirac delta function, whose transforms become simple constants. The transform has applications in physics and engineering, especially in the study of vibrations and transient phenomena in electrical circuits and mechanical structures.[2]

## Contents

* 1 Relationship to the Laplace transform
* 2 Definition
* 3 See also
* 4 References

## Relationship to the Laplace transform

The Laplace–Carson transform V∗(j,p) is directly defined from the one-sided Laplace transform F(s). If F(s)=ℒ{V(j,t)}(s) is the Laplace transform of V(j,t), then the Laplace–Carson transform is given by:

V∗(j,p)=pℒ{V(j,t)}(p)=p∫0∞V(j,t)e−ptdt

where the variable p is used in place of the more common s for the frequency domain.[1] Due to the multiplication by p, the transform of a derivative, ℒ{V′(t)}=pF(p)−V(0), takes a different form that some authors found more convenient for operational calculus.[2]

## Definition

Let V(j,t) be a function and p a complex variable. The Laplace–Carson transform is defined as:[2]

V∗(j,p)=p∫0∞V(j,t)e−ptdt

The inverse Laplace–Carson transform is:

V(j,t)=12πi∫a0−i∞a0+i∞etpV∗(j,p)pdp

where a0 is a real-valued constant chosen so that the integration path lies to the right of all singularities of the integrand.

## See also

* Laplace transform

## References

1. ↑ 1.0 1.1 Debnath, Lokenath; Bhatta, Dambaru (2015). Integral Transforms and Their Applications (3rd ed.). CRC Press. p. 247. ISBN 978-1-4822-2357-6.
2. ↑ 2.0 2.1 2.2 Frýba, Ladislav (1973). Vibration of solids and structures under moving loads.

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