Applications of Non-antidifferentiable FunctionsDate: 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/ |
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