Analytic Function

A function is considered to be analytical if and only if, for any x0 in its domain, its Taylor series about x0 converges to the function in its neighborhood. An analytic function is defined as an infinite differentiation function and it is given by a convergent power series. Almost all the functions that are produced by using fundamental algebraic and arithmetic operations as well as simple transcendental functions, may be described as analytical at any point in their domain.

Types Of Analytic Functions

Analytic functions are of two types:

1. Real analytic function

2. Complex analytic function

Each type of these functions is infinitely differentiable and possesses different properties.

Real Analytic Function

A real analytic function is an infinitely differentiable function whose Taylor series converges to f(x) pointwise for any x in the neighborhood of any point x_0 in its domain.

T(x)=\sum_{n=0}^{\infty}\frac{f^{n}(x_0)}{n!}(x-x_0)^n

A function f(x) is a real analytic function on an open set D in the real number line if for any x0 ϵ D,

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+...

The series is convergent to f(x) for x in the neighborhood of x_0 .

The Collection of all real analytic functions on a set D is denoted by C^w(D) .

Complex Analytic Function

A function is complex analytic if and only if it is holomorphic which requires it to be complex and differentiable.

Let

f(x,y)=u(x,y)+iv(x,y)

be a complex function. Substituting x=(z+\bar z)/2 and y=(z-\bar z)/2i gives

f(z,\bar z)=u(x,y)+iv(x,y)

For f(z,\bar z) to be analytic, a necessary condition is that ∂f∂z=0.

To be analytic f=u+iv should only depend on z. Thus, real and imaginary parts u and v of f must satisfy:

\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}

\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}

The above two equations are the Cauchy-Reimann equations.

For a function to be analytical, the necessary and sufficient conditions are that the partial derivatives of real and imaginary parts \frac{\partial u}{\partial x},\frac{\partial v}{\partial y},\frac{\partial u}{\partial y},\frac{\partial v}{\partial x} must satisfy the Cauchy-Reimann equations and must be continuous.

Properties Of Analytic Functions

• Functions formed by addition, multiplication, or composition of analytic functions are also analytical.

• The limit of uniformly convergent sequences of analytic functions is also an analytic function.

• Analytic functions are infinitely differentiable.

• The function f(z)=\frac{1}{z}, z\neq 0 is analytic.

• The modulus of the function |f(z)| cannot reach its maximum in U if f(z) is an analytic function defined on U.

• If f(z) is analytical and k is a point in its domain then the function \frac{f(z)-f(k)}{z-k} is also an analytic function.

• If f(z) is an analytic function on a disk D, then there is an analytic function F(z) on D such that F’(z) = f(z). F(z) is called the primitive of f(z).

• If f(z) is an analytic function on a disk D, k is a point in the interior of the disk and C is a closed curve that does not pass through k then

W(C,k)=f(k)=\frac{1}{2\pi i}\int C\frac{f(z)-f(k)}{z-k}dz

Where W(C,k) is the winding number of C around z.

• The zeroes of an analytic function are isolated points unless the function is identically zero.

• If C is a curve connecting two points z0 and z1 in the domain in the domain of an analytic function f(z) then,

\int_{c}f^{'}(z)=f(z_1)-f(z_0)

Applications Of Analytic Function

• In mathematical physics, analytic functions are crucial for solving two-dimensional problems.

• Analytic functions are used for fluid flow, electrostatic fields and heat flow problems.

Examples

Following functions are the examples of analytic functions:

• Exponential function

• Hypergeometric functions

• Trigonometric functions

• Bessel functions

• Gamma functions

• Logarithmic functions

• All polynomials