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 exist real numbers and such that for all .

In particular, implies that exists for real .

Carson (1922) used this normalization in interpreting Heaviside's operational calculus. It makes the transform of the Heaviside step function --a function whose Laplace transform is given by --equal to 1 for . 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 within their common domains of convergence, with when division by occurs:

(2)

(3)

and

(4)

For these derivative identities, use ordinary derivatives on and right-hand values at zero. Assume the lower derivatives are locally absolutely continuous, the highest derivative is locally integrable, and the relevant functions have exponential order.

Moreover, one can show that for real and , within their domains of convergence,

(5)

(6)

and

(7)

The identities in (6) and (7) 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, if is locally integrable at zero, 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).