Analytic Functions

In this article, we will define analytic functions in simple words. We will provide you with all the analytic function formulas and examples that will help you to understand this concept easily.  Both real analytic functions and complex analytic functions will be discussed in depth. It is important to understand this concept since it is widely used in many mathematical solutions.

Meaning of Analytic Functions

Analytic functions in simple words are the functions that are mainly derived from a convergent power series. There are two types of analytic functions:

• Real analytic functions
• Complex analytic functions

Both the above functions are infinitely differentiable but complex analytic functions have properties that don’t hold true for the real analytic functions. When a function has derivatives of all the orders then it is called an infinitely differentiable function. 

A function can be analytic only if it satisfies this condition:

• Its Taylor series about x0 converges to the function in some neighbourhood for every x0 in its domain
• At its core, the Taylor series is a function that has an infinite sum of terms expressed at a single point

Real Analytic Functions

A function f is said to be a real analytic function on an open set D in the real line if it satisfies this condition: x0 D

f(x)= n=0 an(x-x0)n= a0+ a1(x-x0 )+a2 (x-x0)2 + a3(x-x0)3+………..

where the coefficients a0 ,  a1  are real numbers and at the same time the series is convergent to f(x) for x in a neighbourhood of x0.

Basically, a real analytic function is an infinitely differentiable function in a way that the Taylor series at any point x0 in its domain:

T(x)= n=0f(n)(x0)n!(x-x0)n

Complex Analytic Functions

A function can only be an analytic function if it is holomorphic i.e. it is complex and differentiable.

Analytic Functions Examples

• Trigonometric functions, logarithm, and power functions are great examples of analytic functions

• Polynomials are also analytic
• If a polynomial has a degree n then the terms that have a degree larger than n in its Taylor series expansion must turn to 0 to turn the series into trivially convergent

• Another example of an analytic function is an exponential function
• The Taylor series for this function must converge for xclose enough to x0 and for all the values of x (real or complex)

Functions that are Not Analytic

• The absolute value function which is present on the real numbers or complex numbers is not analytic everywhere since they cannot be differentiated at 0

• Functions defined by Piecewise (functions which are defined by multiple sub-functions) are also not analytic where the pieces meet

• The complex conjugate function z → z* (a complex number that has an equally real part and at the same time has an imaginary part which is equal in magnitude but opposite in sign) is also not analytic

• Any smooth function f with compact support i.e f C0(Rn) cannot be analytic

Properties of Analytic Functions

• The sums and products of analytic functions are also analytic
• The function f(z)=1/z (z0) is analytic
• The inverse of an analytic number which is nowhere zero is also analytic
• If f(z) and g(z) are analytic functions on, then their sum f(z)+f(g) and their product f(z).f(g) are also analytic
• If f(z) and g(z) are two analytic functions and f(z) is in the domain of g for all z, then their composite g(f(z)) is also an analytic function
• Every non-constant polynomial p(z) has a root which means there exists some z0 such that p(z0) =0

Real and Complex Analytic Functions

Both the real analytic functions and complex analytic functions are infinitely differentiable. Although they are not the same and have their own differences.

According to Liouville’s theorem, the bounded complex analytic function that is defined on the whole complex plane is always constant. Although this statement would not hold true we apply this to real analytic functions, with the complex plane replaced by the real line.

f(x)= 1x2+1

Also, if we define an analytic in complex analysis in comparison with an open ball (solid figure bounded by a sphere) around a point x0 then its power series expansion at x0 is convergent in the whole open ball. Although this statement will not hold true for the real analytic function.

Conclusion

Analytic functions given by a convergent power series are a complex concept. But, we hope that after reading the article you have understood the concept without any complications. 

If you know the Talyor series then you will understand the analytic function concept more clearly. Both the real analytic function and complex analytic function have their own unique properties and applications.