Analytic Function

Introduction

The theory of analytic functions emerged in the nineteenth century, owing primarily to the work of A.L. Cauchy, B. Riemann, and K. Weierstrass. This theory was significantly influenced by the “transition to the complex domain.” The theory of analytic functions was developed as a subset of the theory of functions of a complex variable; currently (in the 1970s), the theory of analytic functions is the primary subject of the general theory of functions of a complex variable.

What is Analytic Function

Analytic Functions are defined in mathematics as functions that are locally given by the convergent power series. Also, we can say For every x0 in its given domain, the function can be called to be analytic if and only if its Taylor series about x0 converges to the given function in some neighbourhood.

 The analytic function is divided into two types: real analytic function and complex analytic function. Real and complex analytic functions are both infinitely differentiable. In general, the complex analytic function has some properties that the real analytic function does not.

Analytic Function Types

Analytic Functions are classified into two types, which are similar in some ways but have distinct characteristics. Analytic functions are classified into two types:

• Real analytic Function 
• Complex Analytic Function

Real Analytic Function

If for any X0  ∈ D, a function “f” is said to be a real analytic function on the open set D in the real line.

Where a0, a1, a2,… are real numbers, and the series is convergent to the function f(x) for x in the neighbourhood of x0.

In other words, the real analytic function is defined as an infinitely differentiable function whose Taylor series converges to the function f(x) for x in a pointwise neighbourhood of x0 at any point x0 in its domain.

Complex Analytic Function

If and only if a function is holomorphic, it is said to be a complex analytic function. It denotes that the function is differentiable in a complex way.

Properties

The following are the fundamental properties of analytic functions:

• An analytic function is also the limit of a uniformly convergent sequence of analytic functions.
• If f(z) and the g(z) are the two given analytic functions on U, then their sum f(z) + g(z) and product f(z) . g(z) is also an analytic function.
• If f(z) and g(z) are the two given analytic functions, and f(z) is in the given domain of g for all z, then their composite g(f(z)) is an analytic function as well.
• The analytic function f(z) = 1/z (z0)
• Constant functions are bounded entire functions.
• There is a root to every nonconstant polynomial p(z). That is, there is some z0 for which p(z0) = 0.
• If f(z) is an analytic function defined on U, then the modulus of the function |f(z)| cannot be maximised in the U.
• Unless f(z) is identically zero, the zeros of an analytic function, say f(z), are the isolated points.
• If F(z) is an analytic function and C is a curve connecting two points z0 and z1 in the given domain of f(z), then C F’(z) = F(z1) – F(z2) (z0)
• If f(z) is an analytic function defined on a disc D, then there is an analytic function F(z) defined on D such that F′(z) = f(z), known as a primitive of f(z), and, as a result, C f(z) dz =0; for any closed curve C in D.
• If f(z) is an analytic function and z0 is any point in f(z domain )’s U, then the function [f(z)-f(z0)]/[z – z0] is analytic on U as well.
• If f(z) is an analytic function on a disc D, z0 is a point inside D, and C is a closed curve that does not pass through z0, then W = (C, z0)f(z0) = (1/2 i)C [f(z)] /[z – z0]dz, where W(C; z0) is the number of times C winds around z.

Why Analytic Function is Important

The class of analytic functions is extremely important for the following reasons:- 

• For starters, the class is sufficiently large; it includes the vast majority of functions encountered in the main problems of mathematics and its applications to science and technology.
• Second, the analytic function class is closed in terms of the fundamental operations of arithmetic, algebra, and analysis. 
• Finally, the uniqueness of the analytic function is a useful property: Each analytic function is an “organically connected whole” that represents a “unique” function throughout its natural domain.

Use of Analytic Function

Analytic functions are crucial in solving two-dimensional problems in mathematical physics. Displacements and stresses in anti-plane or in-plane crack problems can be written as functions of complex potentials.

Eg, The application of a flow equation to an analytic function demonstrates the physical applicability of complex analysis. This work provides a solution to a physical problem in fluid flow by formalising the concept of fluid element motion.

• How Analytic Function used in Oracle

Analytic functions in Oracle are similar to aggregate functions in that they work on a subset of rows and are used to calculate an aggregate value based on a group of rows, but aggregate functions reduce the number of rows returned by the query, whereas aggregate functions do not reduce the number of rows returned by the query after execution.

Point to Remember:

• Analytic functions are functions that are locally given by the convergent power series, according to mathematics. Alternatively, a function is said to be analytic if and only if its Taylor series about x0 converges to the function in some neighbourhood for every x0 in its domain.