This book, divided in ten chapters deals with: algebra of complex numbers and its various geometrical properties, properties of polar form of complex numbers and regions in the complex plane; limit, continuity, differentiability; different kinds of complex valued functions; different types of transformations; conformal mappings of different functions; properties of bilinear and special bilinear transformation; line integrals, their properties and different theorems; sequences and series, Power series, Zero's of functions, residues and residue theorem, meromorphic functions, different kinds of singularities; evaluation of real integrals; analytic continuation, construction of harmonic functions, infinite product, their properties and Gamma function; Schwarz-Christoffel transformations, mapping by multi valued functions, entire functions; and, Jenson's theorem and Poisson-Jenson theorem. The book is designed as a textbook for UG and PG students of science as well as engineering.