Laplace-Carson Transform

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

(1)

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

1. is integrable in every interval of finite length,

2. for all ,

3. There exists a real number such that for all values .

In particular, implies that exists for all real numbers .

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 --a function whose Laplace transform is given by --equal to 1 for all values . Indeed, from the definition of alone, one can easily deduce this property of as well as a collection of other straightforward elementary properties of the transform itself. For example, if is a function whose Laplace-Carson transform is denoted and if is used as shorthand for applying the Laplace-Carson transform to and arriving at , the following identities hold:

(2)

(3)

and

(4)

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

(5)

(6)

and

(7)

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

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

(8)

Finally, one can show that

(9)

and

(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).