The theory of complex variables and complex functions is an essential part of the mathematical toolkit for science and engineering. It plays an important role in the quantitative modeling of waves, oscillations, and other ubiquitous phenomena.This book covers the core mathematical topics of complex algebra and complex analysis, with a strong focus on their notable applications in the physical sciences. Major topics include complex variables, the Cauchy-Riemann equations, contour integration, the calculus of residues, branch cuts, Fourier series, delta functions, Fourier transforms, and Green's functions. The discussion covers many subtopics that are specially relevant in science and engineering, including wave propagation in free space and inhomogenous media, resonances, causality, wavepackets, and Heisenberg's uncertainty principle. In-depth examples explore the practical uses of complex methods in numerous applications, such as modelling the diffraction of light waves.