Analytic functions
Introduction
I was interested in doing my Internal assessment in functions particularly in analytic functions as I was much fascinated with topic (functions)during the study of mathematics during the course. The theory part that includes Taylor series as well as its coverage on complex functionality. In this exploration I surveyed on the theorems associated with analytic functions as well as its functions. This has helped me widen my knowledge and mathematical skills on complex numbers, calcus and functions as topics studied during the course. I have always to kept correct justification on the theories and have used required mathematical models and correct interpretation in the results that I got in the theory processing.
Analytic functions has a very wider application on Analytic modulated system whereas there is a general theory of analytic modulation system and they are developed in the transmitted signal σ(t) = Re {eiwctf(z(t)}.Due to this noticeable physical application in life, it motivated me to write a this maths exploration.

In mathematics, an analytic function could be simply defined as a function that is locally given by a convergent power series. There exist two parts namely real analytic functions and complex analytic functions, functioning differently.
These functions are infinitely differentiable, But as said above functions of each type are infinitely differentiable, having the complex analytic functions exhibiting properties that generally do not hold for real analytic functions.
A function is said to be analytic if and only if its Taylor series about x0 converges to the function for every x0 in its domain.

Having a function w=fz, defined in a domain D, is said to be an analytic function in D if w has a continuous derivative in D. Analytic functions generally have a great applications in physical problems, as a part of mathematics or are derived from such. Lets take an example of taylor series it will be seen that procession of a continuous derivative implies procession of a continuous second derivative, third derivative…

fZo+f'ZoZ-Zo1!+f''Zoz-Zo22!+….

In a neighborhood of each z0 of D. Thus it can be defined as one that is so representable by Taylor series, and often this definition is used.

Symbols and notation
The symbols and mathematical notation f o r the body of the paper a r e defined below Symbols used only in the appendixes are not listed here. Since the appendixes are short

There are several symbols and notation used
Example
Re ( ) –Real part.
L ( )- Linear transformation.
≜ - Equality by definition.
E{ X }- Statistical expectation. - Absolute value. ϕ ( ) – statistical characteristic function. ψ ( ) –complex band pass signal function. λ ( ) – first zone output function of an ideal limiter.

Theorem

If w=u +iv = f(z) is analytic in D, then u and v have continuous first partial derivatives in D and hence it will be satisfying the Cauchy-Riemann equations. Cauchy- Riemann equations are as follows let f(x) be defined in a neighborhood of z0, Recalling that by definition that f is differentiated at z0, with derivative f’(z0), if

smooth f is at x0. The following example shows that this is no longer the case for the complex derivative.

Let f(z) be defined in a neighbourhood of z0. Recall that, by definition, f is differen- tiable at z0 with derivative f′(z0) if lim f(z0 + ∆z) − f(z0) = f′(z0) ∆z→0 ∆z
Whether or not a function of one real variable is differentiable at some x0 depends y on how sh s at x0. The following example shows that this is no