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The present book offers a first course on Complex Analysis. It deals with the differential and integral properties of functions of a single complex variable. It includes conformal transformations and applications of complex analysis to potential theory and flow problems.The subject matter is presented here in five chapters. It starts with a historical note for the introduction and subsequent development of the subject right from the initial stage to the present modern complex analysis. The first chapter includes the introduction of complex numbers, their geometric representation, De Moivre's theorem, roots and logarithm of a complex number and real and imaginary parts of a complex function.The concepts of differential calculus for functions of a complex variable are outlined in the second chapter. It gives the concept of limit and differentiability of a function. Analytic properties and Cauchy-Riemann equations form the main subject matter of this chapter. The third chapter dealswith the integration of functions of a complex variable.