The number is said to be a real number if it s square is greater than or equal to zero. The fundamental theorem of algebra states that every polvuoruial equation of degree n greater than or equal to one has at least one root. The great mathematician Leonhard Euler [1707- 1783] came across a polynomial equation x2 +1 = 0 and found that it has no real roots.

The geometric representation derives its usefulness from the vivid mental pictures associated with a geometric language. We take the point of view, however, that all conclusions in analysis should be derived from the properties of real number's, and from the axioms of geometry.

In classical analytical geometry the equation of a locus is expressed as a relation between x and y. It can just as well be expressed in terms of Z and z, sometimes to distinct advantage. The thing to remember is that a complex equation is ordinarily equivalent to two real equations: in order to obtain a genuine locus these equations should be essentially the same.

In this unit we define functions of complex variables, limit of a complex function and continuity of the complex function. Further we also study certain interesting properties of them.

The objective of this unit to show that a function of a complex variable can have a derivative only if the real and imaginary parts satisfy a system of a partial differential equations called the Cauchy-Riemann equations. Further study of the Cauchy-Riemann equations shows a connect ion between complex analysis and harmonic functions.

In this unit, we introduce the concept of power series and prove that there exists circle of convergence associated with any power series. The power series converges within its circle of convergence, analysis and it s derivative call be obtained by termwise differentiation.

The person who approaches calculus exclusively from the point of view of real numbers will not expect any relationship between the exponential function and the trignometric functions cos x and sin x. Indeed, these functions seem to be derived from completely different sources and with different purposes in mind.

In this unit we introduce the concept s of topology of complex numbers and geometry of t he complex valued functions of a complex variable such as linear transformation, elementary conformal mappings.

The integration of a function of a complex variable is carried out over a curve and leads to results of great importance in pure and applied mathematics. In this unit, we discuss first complex valued functions then curves and finally integration over curves.

There are several forms of Cauchy's theorem, but they differ in t heir topological rather than in their analytical content. It is natural to begin with a case in which the topological considerations are trivial. In this unit we determine Cauchy theorem for a rectangle and Cauchy' s theorem for a disc.

In this unit, we establish another fundamental result known as Cauchy's integral formula using Cauchy's theorem. As application of Cauchy's integral formula. we also deduce Morera's theorem, Liouville's theorem and Fundamental theorem of Algebra.

This unit is devoted to a consideration of functions which are analytic at all points in a bounded domain except at a finite number of points. Such exceptional points are known as singular points. In this content, the first portion will naturally consists of classification of singular points in terms of the behavior of the function in their neighborhood.

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