This book is rigorous introduction to the theory of functions of complex of one variable. An effort has been made to choose those topics that appeal to most students in either in pure mathematics or applied mathematics. This is a book for a first course in functions of complex variable, assuming a knowledge of freshman calculus. It is designed for students in engineering, physics, and mathematics. Without sacrificing ease and clarity of proofs, mathematical preciseness and rigor are stressed. Cross references are used to justify almost every step in each proof and solutions.