LAPLACE TRANSFORMATION

The Laplace transform of a function f(t), defined for all real numbers t ≥ 0, is the function F(s), defined by:

The parameter s is a complex number:

with real numbers σ and ω.

Other notations for the Laplace transform includeor alternativelyinstead of F.

The meaning of the integral depends on types of functions of interest. A necessary condition for existence of the integral is that f must be locally integrable on [0,∞). For locally integrable functions that decay at infinity or are of exponential type , the integral can be understood as a (proper) Lebesgue integral. However, for many applications it is necessary to regard it as a conditionally convergent improper integral at ∞. Still more generally, the integral can be understood in a weak sense, and this is dealt with below.

One can define the Laplace transform of a finite Borel measure μ by the Lebesgue integral.

An important special case is where μ is a probability measure or, even more specifically, the Dirac delta function. In operational calculus, the Laplace transform of a measure is often treated as though the measure came from a distribution function f. In that case, to avoid potential confusion, one often writes

where the lower limit of 0⁻ is shorthand notation for

This limit emphasizes that any point mass located at 0 is entirely captured by the Laplace transform. Although with the Lebesgue integral, it is not necessary to take such a limit, it does appear more naturally in connection with the Laplace transform.

Probability theorem

the Laplace transform is defined as an expected value . If X is a random variable with probability density function f, then the Laplace transform of  f is given by the expectation

 this is referred to as the Laplace transform of the random variable X itself. Replacing s by −t gives the moment generating function of X. The Laplace transform has applications throughout probability theory, including first passage time of stochastic processes such As Markov chains , and renewal theory.

Of particular use is the ability to recover the cumulative distribution function of a random variable X by means of the Laplace transform as follow: