Functions of a complex variable are functions of (x, y) that depend exclusively on the combination (x + iy), and functions of this type that may be extended in power series in this variable are of special importance. This combination (x + iy) is known as z, and we may write functions like zn, exp(z), sin z, and all the other common z and x functions.

They are defined identically, with the exception that they are complex-valued functions, i.e., vectors in this two-dimensional complex number space, each containing a real and imaginary portion (or component). The values of the majority of the standard functions we've covered so far are real when their arguments are real. The square root function, which becomes imaginary with negative parameters, is the obvious exception.

We may build any polynomial in z, as well as any power series, by multiplying z by itself and by any other complex number. The power series expansions of the exponential and sine functions of z converge everywhere in the complex plane and are used to define them. Because all of the procedures used to create standard functions can also be used to create complex functions, we can create all of the standard functions of a complex variable using the same techniques used to create standard functions for real variables.

In this “Function of Complex Variables - Mathematics II” you will learn about the following topics:

- Introduction to Function of Complex
Variables
- Complex Variable
- The function of Complex Variables
- Analytic Function
- Necessary and Sufficient
Conditions for F (Z) To Be Analytic (Without Proof)
- Harmonic Function
- Conformal Mappings

==== Point to Note ====

This article Function of Complex Variables - Mathematics II is contributed by Namrata Chaudhary, a student of Lumbini Engineering College (LEC).

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