Analytic Functions - Complex Analysis, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

Definitions and Theorems

Definition 

A function f (z) is said to be analytic in a region R of the complex plane if f (z) has a derivative at each point of R and if f (z) is single valued.

Definition 

A function f (z) is said to be analytic at a point z if z is an interior point of some region where f (z) is analytic.

Hence the concept of analytic function at a point implies that the function is analytic in some circle with center at this point.

Theorem

If f (z ) is analytic at a point z , then the derivative f 0 (z ) is continuous at z .

Corollary

If f (z) is analytic at a point z , then f (z) has continuous derivatives of al l order at the point z.

Conditions for a Complex Function to be Analytic

A necessary condition for a complex function to be analytic

Let

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

be a complex function. Since x = (z + )=2 and y = (z ¡ )=2i, substituting for x and y gives 

f (z; ) = u(x; y) + iv(x; y)

A necessary condition for f (z ,) to be analytic is

                              (1)

Therefore a necessary condition for f = u + iv to be analytic is that f depends only on z . In terms of the of the real and imaginary parts u; v of f , condition (1) is equivalent to                     (2)

                     (3)

Equations (2, 3) are known as the Cauchy-Riemann equations. They are a necessary condition for f = u + iv to be analytic.

Necessary and sufficient conditions for a function to be analytic

The necessary and sufficient conditions for a function f = u + iv to be analytic are that:

1. The four partial derivatives of its real and imaginary parts   satisfy the Cauchy-Riemann equations (2, 3).

2. The four partial derivatives of its real and imaginary parts  are continuous.

Theorem

If f (z ) is analytic, then

                        (4)

                       (5)

The real and imaginary parts of an analytic function are harmonic conjugate functions, i.e., solutions to Laplace equation and satisfy the Cauchy Riemann equations (2, 3).

Singularities of Analytic Functions

Points at which a function f (z) is not analytic are called singular points or singularities of f (z). There are two different types of singular points:

Isolated Singular Points

If f (z) is analytic everywhere throughout some neighborhood of a point z = a, say inside a circle C : |z - a| = R, except at the point z = a itself, then z = a is called an isolated singular point of f (z). f (z) cannot be bounded near an isolated singular point.

Poles

If f (z ) has an isolated singular point at z = a, i.e., f (z) is not finite at z = a, and if in addition there exists an integer n such that the product

(z - a)ⁿ f (z)

is analytic at z = a, then f (z) has a pole of order n at z = a, if n is the smallest such integer. Note that because (z - a)ⁿ f (z) is analytic at z = a, such a singularity is called a removable singularity. Example: f (z) = 1=z² has a pole of order 2 at z = 0.

Essential Singularities

An isolated singular point which is not a pole (removable singularity) is called an essential singular point. Example: f (z) = sin(1/z) has an essential singularity at z = 0.

Branch Points

When f (z) is a multivalued function, any point which cannot be an interior point of the region of definition of a single-valued branch of f (z) is a singular branch point. Example: f (z) =  has a branch point at z = a.