Complex analysis is a branch of mathematical analysis that studies functions of complex numbers. It involves exploring properties of complex-valued functions, including differentiation, integration, and series expansions. Key concepts include analytic functions, which are differentiable in a complex sense, and the Cauchy-Riemann equations, which provide conditions for differentiability. Complex analysis also deals with contour integration, residue theory, and the study of singularities. It has applications in various fields like engineering, physics, and number theory.