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Unformatted text preview: C HEMICAL E NGINEERING 132A Professor Todd Squires Spring Quarter, 2007 Review for Final FOURIER ANALYSIS Remember, the basic idea of Fourier Analysis is to take a function and represent it in a different form. It is all the same information, just a different way to present it. I had used an analogy with language: if you wanted to read Homers Iliad, you have two ways to do it: either learn ancient Greek, or buy a translation. In a variety of situations in Mathematics/Physics/Engineering, it turns out to be far easier to transform, or translate functions than to use the standard f ( x ) form. We have already seen this with the heat / diffusion equation we very naturally wind up with a solution to the PDE in the form of a Fourier series, or a Fourier-Bessel Series. There are a variety of transforms we have used. In the first half of class, we used Laplace Transforms, which we used primarily to solve initial value ODEs. The second half of class saw Fourier Series, Fourier Sine and Cosine Series, Fourier Transforms, and Fourier-Bessel Series. The basic idea of the Fourier Series and its variants is to take a function and to represent it as a combination of sines and cosines oscillatory (wiggly) functions, each with a frequency or wavenumber q . High q means rapid oscillations, low- q means slow oscillations, and q = 0 means a constant. The cochlea in your ear is a natural Fourier analyser it takes a time-dependent pressure signal and converts it into sound frequencies, each with some amplitude. X-Ray crystallography gives Fourier-transformed structures. As we will see in a bit, the Fourier-Bessel Series is morally very similar to the Fourier Series the difference being that Bessel Functions J n ( qx ) and Y n ( qx ) are used in place of sin( qx ) and cos( qx ). Bessel Functions are also wiggly functions, and q functions a lot like a frequency/wavenumber, just like it does for sines and cosines. Note also the subscript n this is called the order of the Bessel function. Well talk about this in a bit. The short version is that Bessel functions with different n s should be thought of as totally different functions. (They are linearly independent, for those who know what that means). The simplest way to start is with the Fourier Series, which gives a way to represent a function on a finite domain, defined between L and L . This arises in many contexts we used them in the one-dimensional heat/diffusion equation (temperature in a thin bar of length 2 L , for example). The Fourier series representation of a function f ( x ) is f ( x ) = A 2 + summationdisplay n =1 A n cos nx L + B n sin nx L . (1) There are several things to notice here....
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