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.