Fourier Series. This was the super annoying memorizing-the-formulas topic in my class, but it was actually one of the most interesting math concepts ever. So what is the Fourier series? What's the significance of it? For that we first have to understand what a periodic function is. A periodic function is a function where T > 0, and f(x+T) = f(x) for every value of x. The T is the period of f(x). An example of a periodic function is sin(x) and cos(x), which have a period of 2π. Fourier series is essentially a way to expand this periodic function to an infinite series involving a bunch of sines and cosines. So how do you derive the Fourier series of a periodic function? Let p > 0 and f(x) be a periodic function with period 2p, within the bounds of (-p, p). The Fourier series of f(x) is: where the a of n and a of 0 and b of n are Fourier coefficients ALSO, there are two things to keep in mind: assuming x is an integer,
the left graph shows the original f(x) and the right graph shows Fourier series estimation graph. When you graph the certain number of n terms of the fourier series, you will get a close approximation. This series is very similar to Taylor series, except Fourier series also works with discontinuous functions as well. Fourier sine seriesYou can obtain the Fourier sine and cosine series from the general formula. For the Fourier Sine Series, we assume that f(x) is an odd function, which means that f(-x) = -f(x). If thats the case, then the a of 0 and a of n terms become zero because an odd function (f(x)) multiplied by an even function (cos(nπx)) = an odd function. An interval from -p to p over an odd function is 0. an example of an odd function over an interval (-p, p). both the areas cancel each other out, which evaluates the integral to 0. Since a of n and a of 0 are both equal to 0, the function evaluates to the general Fourier series with just the b of n Fourier coefficient. Fourier cosine seriesYou obtain the Fourier cosine series when you assume that f(x) is an even function; which means that f(-x) = f(x). Since f(x) is an even function, we know that the b of n term in the general Fourier series equation is 0 because an even function (f(x)) times an odd function (sin(nπx)) is equal to an odd function. The integral from (from -p to p) of an odd function is always zero since the areas cancel each other. Therefore, with only the a of n an a of 0 terms, the Fourier series becomes the Fourier cosine series (with only cosines).
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