Introduction to Complex analysis

complex analaysis  is a branch of mathematics that deals with complex numbers, their functions, and their calculus. In simple terms, complex analysis is an extension of the calculus of real numbers to the complex domain. We will extend the notions of continuity, derivatives, and integrals, familiar from calculus to the case of complex functions of a complex variable. In doing so we will come across analytic functions, which form the centerpiece of this introduction. In fact, to a large extent complex analysis is the study of analytic functions.

The basic ingredient of complex analysis is an analytic function, or that we know so well in calculus as a differentiable function. Any complex number z can be thought of as a point in a plane (x,y), so z = x+iy, where i=√-1.  In a similar fashion, any complex function of a complex variable z can be separated into two functions, as in,  f(z)=u(z)+iv(z), or, f(x,y)=u(x,y)+iv(x,y). Clearly, such functions depend on two independent variables and have two separable functions, so plotting the function would need a four-dimensional space, which is difficult to imagine.

Of course, the first starting point of the calculus of complex functions is to start with continuity of the function and then slowly move into the differentiability in the complex domain. 

Continuity 

We start with a rather trivial case of a complex-valued function. Suppose that f is a complex-valued function of a real variable. That means that if x is a real number, f(x) is a complex number, which can be decomposed into its real and imaginary parts:  f(x) = u(x)+i v(x), where u and v are real-valued functions of a real variable; that is, the objects you are familiar with from real calculus. We say that f is continuous at x0 if u and v are continuous at x0.

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