The Laplace transform presents an alternative method of solving differential equations. It enables us to transform a differential equation into an algebraic equation. We can solve the algebraic equation and apply an inverse transformation to obtain the solution to the differential equation. The Laplace transform also enables us to solve initial value or boundary value problems, both homogeneous and non homogeneous.
We can apply the Laplace transform to any function that is piecewise continuous on
and such that
for some constants
and all
for some
The Laplace transform works because of the factor
This is a decreasing function of
so the integral converges for any function of
for which
does not increase 'too fast'. A very wide range of functions satisfies this criterion.
The Laplace transform is related to the Fourier transform, but whereas the Fourier transform resolves a function into components of frequency, the Laplace transform resolves a function into its moments. Like the Fourier transform, the Laplace transform is used for solving differential and integral equations. Given a simple mathematical or functional description of an input or output to a system, the Laplace transform provides an alternative functional description that often simplifies the process of analysing the behaviour of the system.
The Laplace transform, denoted
is a linear operator on a functionwith a real argument
that transforms it to a function
with a complex argument
This transformation is essentially bijective, with respective pairs ofand
matched in tables. The Laplace transform has the useful property that many relationships and operations over the original correspond to simpler relationships and operations over
The Laplace transform of a functionis defined as
where
is in general a complex number. The transformed function is a function of
labelled
Example: Find the Laplace transform of
Example: Find the Laplace transform of
