Fourier Analysis
Series and transforms: decomposing functions into pure frequencies.
A $2\pi$-periodic, square-integrable function has a Fourier series
$$f(x) = \sum_{n \in \mathbb Z} c_n e^{inx}, \quad c_n = \frac{1}{2\pi}\int_{-\pi}^\pi f(x) e^{-inx}\, dx.$$Convergence is in $L^2$; pointwise convergence requires more (Dirichlet, Carleson). For non-periodic $f \in L^1(\mathbb R)$ define the Fourier transform
$$\hat f(k) = \int_{-\infty}^\infty f(x) e^{-ikx}\, dx, \qquad f(x) = \frac{1}{2\pi}\int \hat f(k) e^{ikx}\, dk.$$Key properties: $\widehat{f'} = ik \hat f$ (derivatives become multiplications), $\widehat{f * g} = \hat f \cdot \hat g$ (convolution = product). Parseval/Plancherel:
$$\int |f(x)|^2\, dx = \frac{1}{2\pi} \int |\hat f(k)|^2\, dk.$$Heisenberg's uncertainty principle in QM is a special case: tight $f$ ⇔ broad $\hat f$.