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Laplace-Carson Transform


The Laplace-Carson transform F of a real-valued function f is an integral transform defined by the formula

 F(p)=pint_0^inftye^(-pt)f(t)dt.
(1)

In most cases, the function F is defined only for certain functions f which lie in a class L(f) of real-valued functions. Functions in L(f) satisfy three properties, namely:

1. f(t) is integrable in every interval I subset R of finite length,

2. f(t)=0 for all t<0,

3. There exists a real number c>0 such that |f(t)e^(-ct)|<0 for all values t>=0.

In particular, f in L(f) implies that F exists for all real numbers p in R.

One may regard the Laplace-Carson Transform as a variation of the regular Laplace transform specifically devised by Carson to make the transform of the Heaviside step function H(t)=H_0(t)--a function whose Laplace transform is given by L[H(t)](p)=1/p--equal to 1 for all values p. Indeed, from the definition of F alone, one can easily deduce this property of H(t) as well as a collection of other straightforward elementary properties of the transform itself. For example, if f is a function whose Laplace-Carson transform is denoted F and if f(t)->F(p) is used as shorthand for applying the Laplace-Carson transform to f and arriving at F, the following identities hold:

 int_0^tf(tau)dtau->1/pF,
(2)
 (df)/(dt)->pF-pf(0),
(3)

and

 (d^n)/(dt^n)f->p^nF-sum_(k=0)^(n-1)p^(n-k)(d^k)/(dt^k)f(0).
(4)

Moreover, one can show that for arbitrary real numbers alpha and beta,

 f(t/alpha)->F(alphap),
(5)
 f(t-alpha)H(t-alpha)->e^(-alphap)F(p),
(6)

and

 e^(-betat)->p/(p+beta)F(p+beta).
(7)

The identities in () and () are known as the lag and displacement theorems, respectively.

Given functions f,psi whose Laplace-Carson transforms are F,Psi, respectively, one can show the convolution/multiplication theorem:

 int_0^tf(tau)psi(t-tau)dtau->1/pF(p)Psi(p).
(8)

Finally, one can show that

 (f(t))/t->pint_p^infty(F(q))/qdq
(9)

and

 -tf(t)->pd/(dp)((F(p))/p).
(10)

In addition to the above, one can prove a number of more colorful results about the Laplace-Carson identity using various other methods; many such results require a bit more sophistication (Rubinstein and Rubinstein 1999).


See also

Convolution, Heaviside Step Function, Integrable, Integral Transform, Laplace Transform

This entry contributed by Christopher Stover

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References

Rubinstein, I. and Rubinstein, L. Partial Differential Equations in Classical Mathematical Physics. Cambridge, England: Cambridge University Press, 1999.

Cite this as:

Stover, Christopher. "Laplace-Carson Transform." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Laplace-CarsonTransform.html

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