Applications of Non-antidifferentiable Functions

Date: 01/29/99 at 10:11:06
From: mekisha parks
Subject: Applications of Non-antidifferentiable Functions

What are the real life applications of the functions sin (x)/x, 
e^(-x^2), and sin (x^2)?

Your help would be greatly appreciated.
 
Thank you in advance.

Date: 02/02/99 at 12:19:54
From: Doctor Nick
Subject: Re: Applications of Non-antidifferentiable Functions

The curve y = e^(-x^2) is a commonly known as the "normal curve" or 
"bell curve." Its applications are everywhere. Take a look at any book 
on statistics for lots of examples. Usually the curve doesn't appear as 
simply as just e^(-x^2), but e^(-x^2) is the prototype for every normal 
curve.

The function sin(x)/x is an oscillating function that decays to zero as 
x goes to infinity. Much oscillating behavior behaves this way; it is 
generally known as damped oscillation. If you pluck a string on a 
guitar and let it vibrate, the sound coming from it can be represented 
as a sine wave times a "damping factor" g(t). The damping factor is 
there to account for the fact that the guitar string doesn't vibrate 
forever, but gets quieter and quieter as time goes by. The damping 
factor g(t) has the property that it goes to zero as t goes to 
infinity. The simplest g(t) we could choose here to model the guitar 
sound is with g(t)=1/t. This gives us sin(t)/t as the wave. This may 
not model the wave precisely, but it's definitely a starting point: we 
could try it, compare it with data, and go from there.

I can't think of any "real-life" applications of sin(x^2). However, 
there are many natural phenomena that behave like sin x, and x^2, and 
it's reasonable to think that there must be something out there that 
behaves like the composition of these two functions (i.e. like 
sin(x^2), or (sin x)^2). Perhaps you can be the first to discover such 
a thing, if one has not been discovered yet.

I hope this helps.  

- Doctor Nick, The Math Forum
  http://mathforum.org/dr.math/   

Date: 07/26/2001 at 08:42:13
From: Doctor Rick
Subject: Sine wave damping

I'd like to add something to what Dr. Nick said about the function 
sin(x)/x. In general shape, its graph resembles the decaying oscillations 
of a plucked string, but as he says, it "may not model the wave precisely." 
In fact, the damped oscillations that occur in physical systems such as 
the plucked string are generally of the form

  f(t) = A*e^(-bt)*sin(2*pi*t/T-phi)

This function appears frequently in physics, engineering, etc., because 
its derivatives have the same form, with a change in amplitude and phase. 
Thus it is definitely not among the non-integrable functions in which you 
are interested.

The function sin(x)/x is quite different. It appears frequently enough in 
signal and image processing that it has its own name: the sinc function. 
More precisely, the sinc function is generally defined as

  sinc(x) = sin(pi*x)/(pi*x)

Its usefulness arises because it is the Fourier transform of a rectangular 
pulse. A rectangular pulse is a signal that starts at zero, then sharply 
rises to some value (say, 1), stays there a while, then falls sharply back 
to zero. If you fed this signal into a frequency analyzer and plotted the 
frequency spectrum of the signal, you'd see the sinc function.

Conversely, suppose you had a signal consisting of a single sharp "blip". 
The frequency spectrum of this blip is very broad. Pass it through an ideal 
low-pass filter to keep only those frequencies below a sharp cut-off 
frequency. As the signal comes out of the filter, view it with your frequency 
analyzer, and it will look the way a rectangular pulse looks on an 
oscilloscope. View it on an oscilloscope, and you'll see the sinc function.

Here is one Web site that shows the rectangular pulse and its Fourier 
transform, the sinc function.

  Time-Frequency Relationships - John Glover, University of Houston
  http://www2.egr.uh.edu/~glover/applets/Fourier/TimeFrequency.html   
  
- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/