10: Fourier Series and Fourier Transforms
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The Fourier transform is one of the most important mathematical tools used for analyzing functions. Given an arbitrary function f(x), with a real domain (x∈R), we can express it as a linear combination of complex waves. The coefficients of the linear combination form a complex counterpart function, F(k), defined in a wave-number domain (k∈R). It turns out that F is often much easier to deal with than f; in particular, differential equations for f can often be reduced to algebraic equations for F, which are much easier to solve.