**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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Tags: Integral Transform, Laplace Transform, Math, Monday Math

This entry was posted on December 7, 2009 at 12:27 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed.
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February 8, 2010 at 4:03 am |

[…] 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 […]

February 13, 2010 at 12:14 pm |

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