## 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).

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### 2 Responses to “Monday Math 100”

1. Monday Math 107 « Twisted One 151's Weblog Says:

[…] respectively. What, then, might we say about the product of those transforms, F(s)G(s)? From the definition of the Laplace transform, we see and . So then the product is . Renaming the variables of integration from t to u and v, we […]

2. Old Engineer Says:

So, were you going to say some more about the Laplace Transform? I think I may be missing the point of this post.