None of the regular scheduled posts until 2010.
Archive for December, 2009
The Laplace Transform
Part 2: Convergence
When considering the Laplace transform
we must consider the region of convergence; those values of the complex number s for which the above integral converges.
A function f(t) is of exponential order at infinity if and only if there exist real numbers M and a, and positive real number T such that for all t>T; note then that
Thus, we see that if the function f(t), t≥0, is piecewise continuous and has exponential order at infinity with , then the Laplace transform is defined for .
In fact, it can be shown that if the Laplace transform converges for some , then it converges for any s with , and thus the region of convergence is (if not the entire complex plane) a half-plane of the form , with all or some of the boundary possibly included.
If one punctures a small hole in the side of a large reservoir of liquid, at a point below the surface level of the liquid, one will have a jet of liquid from the hole. The lower the hole, the faster this jet will be. If the liquid in the reservoir has a depth of H, at what height h above the bottom of the reservoir should one place the hole so that the jet gets the most range?
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).
Consider a spool of string with radius r, and ends capped by disks of radius R>r. The spool is placed on its side on a horizontal, so that it can roll on end rims. Suppose one pulls on an unwound end of the string, at some angle θ from the underside horizontal, as pictured below.
It’s clear from basic physical intuition that if θ=90°, so that one pulls straight up, the spool will tend to unwind, rolling to the left in our picture.
Similarly, if we pull horizontally, θ=0°, then the spool will roll to the right, and will wind up the string.
So, then, somewhere between these two angles should be a transition between these behaviors, a critical angle θc where the spool drags without rotating, neither winding nor unwinding. What is this angle?