An analytic function is given locally by a convergent power series. Both real analytic and complex analytic functions are differentiable infinitely, but there are some properties of complex analytic functions that holds true which are not possessed by real analytic ones. A function ‘f’ which is defined on an open subset ‘U’ of ‘R’ or ‘C’ is called analytic if and only if it is given by a convergent power series locally.

A function is analytic only if the Taylor series associated to it about ‘x0’ is converging to the function in any neighborhood for every ‘x0’ to be in its domain.
Let a function ‘f’ be real analytic on an open set say ‘D’ that lies on the real line. Then for any ‘x0’ in ‘D’, we can write as,
 $f(x)=\sum_{n=0}^{\infty }a_{n}(x-x_{0})^{n} = a_{0}+a_{1}(x-x_{0})+a_{2}(x-x_{0})^{2}+...$
Here, the coefficients a0, a1,….., belong to the set of real numbers. Also this series will converge to some f(x) for ‘x’ lying in the neighborhood of ‘x0’.
An analytic function is also a differentiable function which is infinitely differentiable alternatively, in such manner that its domain at any particular point ‘x0’ will converge to some f(x) at certain ‘x’ which lies in the neighborhood of ‘x0’ uniformly.
$f(x)$=$\sum_{n=0}^{\infty }$$\frac{f^{(n)}x_{0}}{n!}$$(x-x_{0})^{n}$Generally, the set of all real valued such functions are denoted by CW(D), on a given set ‘D’. We can define a function ‘f’ on a subset of the real line to be a real analytic one at some point ‘x’ only if there exist some neighborhood ‘D’ of ‘x’ in which ‘f’ is real and analytic.

Also, if ‘f’ is a differentiable equation which is infinitely differentiable on an open subset ‘D’ contained in ‘R’, then the conditions stated below are all equivalent.
1).  Function ‘f’ is analytically real.
2).  There will exist an extension of ‘f’ which is complex to an open set ‘G’ contained in ‘C’ which will contain ‘D’.
3).  Let us have a compact set ‘K’ contained in ‘D’. Then there exists a constant ‘C’ in a manner that for each ‘x’ belonging to K and for every ‘k’ (non negative integer), the given bound holds true. 
$|\frac{d^{k}f(x) }{dx^{k}}|$ $\leq C^{k+1}k!$The FBI transform can be used to characterize the real analyticity of function ‘f’ for a given point say ‘x’.

While complex analytic ones being completely equivalent to holomorphic functions can be characterized much more easily and simply.

 Analytic Function

We will discuss a common application of a function which is analytic that is complex functions.
Complex Analytic Functions: 
The simplest of all the complex functions that are analytic are rational functions.
$R(z)$= $\frac{P(z)}{Q(z)}$
Here, P(z) = am zm + am-1 zm-1 + ⋯ + a1 z + a0 and Q(z) = bn zn + bn-1 zn-1 + ⋯ + b1 z + b0
Also, P(z) and Q(z) are polynomials with complex coefficients am, am-1, …, a1, a0, bn, bn-1,…, b1, b0.
We assume that am and bn, both are not equal to zero. We also make assumption that P(z) and Q(z) does not have any factors in common.
The function R(z) is differentiable at every point of $C\rightarrow S$, and S is the finite set of points within C at which the denominator Q(z) vanishes. The maximum of the two degrees from among the degree of the numerator and the degree of the denominator, is termed as the degree of the rational function R(z).
The sum of two rational functions is also rational function, as is the product. Also, one can check that with these operations of addition and multiplication, the space of rational functions satisfies the field axioms, and this field is denoted by C(X). This is an important example of a field.
The Fundamental Theorem of Algebra allows us to factor the polynomials P(z) and Q(z),
$P(Z)= a_{m}(z-z_{1})^{p1}(z-z_{2})^{p2}....(z-z_{r})^{pr}$

and $Q(Z)= b_{n}(z-s_{1})^{q1}(z-s_{2})^{q2}....(z-s_{r})^{qr}$

Where the exponents denote the multiplicities of the roots, and this provides us with the first of the two important canonical forms for a rational function:

$R(Z)$=$\frac{a_{m}(z-z_{1})^{p1}(z-z_{2})^{p2}....(z-z_{r})^{pr}}{ b_{n}(z-s_{1})^{q1}(z-s_{2})^{q2}....(z-s_{r})^{qr}}$

The zeros z1,…., zr of the numerator are called the zeros of the rational function, while the zeros s1,…, sr of the denominator are called its poles. The order of a zero or pole is its multiplicity as a root of either the numerator or the denominator. Poles of order one are said to be simple.

We say that a rational function R(z) is proper if
m = deg P(z) $\leq$ n = deg⁡ Q(z)
and strictly proper if m < n. If R(z) is not strictly proper, we can divide by the denominator to obtain
R(z)= P1 (z) + R1 (z),
where P1(z) is a polynomial and the remainder R1(z) is a strictly proper rational function.

Analyticity with Differentiability
We know that any function that is analytic may it be real or complex in differentiable infinitely which is also termed as being smooth, or $C^{\infty }$. This differentiability is for the case of the real variables.
When we consider complex functions which are analytic and complex derivatives the situation is much different. We can easily prove that in an open set any complex function which is differentiable in the complex manner is analytic. Also, in case of analyzing complex system, holomorphic functions and functions that are analytic are synonymous or alike in other words.
Some common examples of such functions are: 
1). All polynomials may it be real or complex. This is so because for a polynomial of degree (highest power) ‘n’, the terms of degree greater than ‘n’ in its corresponding Taylor series expansion will immediately merge to 0 and thus the series will be convergent trivially. Also, to add every polynomial is itself is Maclaurin series.

2). All exponential functions are also analytic. This is because all Taylor series for such functions will converge for all values may be real or complex of ‘x’ very closely to ‘x0’, as in the definition.

3). For any open set in the respective domains, the trigonometric functions, power functions and logarithmic functions are also analytic.
 Example of Analytic Function

Strictly proper rational functions also have a second canonical form which is termed as the partial fraction decomposition which is very useful in the calculation of integrals of various rational functions, as it is seen in calculus.

Example: Find out the possible values of
i-2i = exp ?(-2i log?(i))

Solution: For finding possible values of the given function, we first see to it that,
log?(i) = log ?1 + i arg  ?[ Because (i) = 0 + i $\frac{pi}{2}$ + 2$\pi$ki, for every k that belongs to integer set
This gives, i-2i = exp? ( $\pi$ + 4$\pi$k), for every k belonging to the set of integers]
This example shows us that the complex number z$\alpha$, could also possibly have different values infinitely similar to logarithms. Even if square root functions can only have at max two values, then too they are a good example of multi valued functions. 
Analytic functions properties are as follow:
1). The compositions, sums or products of the functions that are analytic are also analytic.

2). For a function that is analytic, its reciprocal if it is not at all zero, is also analytic. Also, the inverse of a function that is analytic derivative of which should not be zero is again analytic.

3). A function that is analytic is infinitely differentiable. In other words we can say that an analytic function is smooth. But the converse of this statement is not true, that is all infinitely differentiable functions are not analytic. This is because in some manner, the functions that are analytic are sparse when compared to all the infinitely differentiable functions.