Monday, April 25, 2005

Power Series, Fourier Series, Laurent Series

In my complex analysis class we just finished the Fourier series. Well fourier series are remarkable. They are just Trigonometric series and thier usefullness in solving the classic problems of heat and wave was fascinating to learn. We learned how to find Fourier coefficients of the series. Its easy to understand because integration between -L to L for combination of sin(nPix/L) and cos(nPix/L) gives you either zero or L and you can have sine series and cosine series. Also if the function is in fourier series its easy to find the fourier coefficients. Dr. Grimmer explained it so well in the class. Now we are taking a look at the Power series. Now these are interesting. It was interesting to see that any polynomial can be expressed in terms of power series at different points. For example the ploynomial x can be reperesented as x only at point 0. But becomes 1+(x-1) at 1 and 2+(x-2) at 2 and so on. Well power series work for only positive powers and not fraction. So what we do if we have to find negative exponents and then you get what is called Laurent series. Still Laurent series don't have a fractional powers. One good thing about power series is that inside their radius of convergence they are differentiable and hence continious.


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