10: Fourier Series and Fourier Transforms

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 (\(x \in \mathbb{R}\)), 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 (\(k \in \mathbb{R}\)). 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.

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