## Monday Math 100

The Laplace Transform
Part 1: Introduction

One of the oldest and best known integral transforms is the Laplace transform; it and the related Fourier transform are likely the most commonly used integral transforms. Like the Fourier transform, it transforms a function from one independent variable, the “time domain”, to a second independent variable, corresponding to a sort of “complex frequency.” It has applications in a number of areas, including differential equations, integral equations, and probability theory.

Specifically, for a function f(t), defined for real t≥0, the Laplace transform (more specifically, the “unilateral” or “one-sided Laplace transform) is the function F(s) defined by
,
where s is a complex number.

From the linearity of the integral, we see that the Laplace transform is linear; for functions f(t) and g(t), and constants a and b,
.

The Laplace transform is “unique” in that if two functions f(t) and g(t) have the same Laplace transform,
,
then  for nonnegative t, except possibly at a finite number of separate points.
(More specifically, if , then  for all a>0).