This course provides a comprehensive introduction to complex analysis, a cornerstone of higher mathematics with profound applications in engineering, physics, and mathematical sciences. You will explore functions of a complex variable, the foundational theory behind analytic and harmonic functions, contour integration and the Cauchy integral theorem, power and Laurent series, calculus of residues, and advanced mapping techniques including conformal mappings. The course culminates in the study of Fourier transforms and their use in solving partial differential equations, equipping you with analytical tools used in real-world scientific modelling.