Residue Theorem - 2 - Free MCQ Practice Test with solutions, GATE MA Engineering

Concept:

Pole:

The value for which f(z) fails to exists i.e. the value at which the denominator of the function f(z) = 0.

When the order of a pole is 1, it is known as a simple pole.

Residue:

If f(z) has a simple pole at z = a, then

If f(z) has a pole of order n at z = a, then

Calculate:
Given:

For calculating pole:

z² + a² = 0

∴ (z + ia)(z - ai) = 0

∴ z = ai, -ai.

∴ z has simple pole at z = ai and -ai.

Residue:

If f(z) has a simple pole at z = a, then

For pole at z = ai

For pole at z = -ai

∴ z has a simple pole at z = ai and  is a residue at z = ia of f(z)

Cauchy's Residue Theorem:

Residue of f(z):

Residue of f(z) is denoted as Res[f(z) : z = z₀]

z₀ is a simple pole of the function f(z)

If f(z) = p(z) / q(z)

Where, p(z), q(z) are polynomials

Then residue is,

Res[f(z) : z = z₀] = 
If f(z) has a pole of order 'm' at z = z₀ then

Res [f(z) : z = z₀] =
Calculation:

Given,

Pole of f(z) has order "2" 

Residue Theorem:

If f(z) is analytic in a closed curve C except at a finite number of singular points within C, then

⁡f(z)dz = 2πi ×  [sum of residues at the singualr points with in C]

Formula to find residue:
1. If f(z) has a simple pole at z = a, then

2. If f(z) has a pole of order n at z = a, then

Calculation:

Concept:

Residue Theorem: 

If f(z) is analytic in a closed curve C except at a finite number of singular points within C, then

∫cf(z) dz = 2πj × [sum of residues at the singular points within C]

Formula to find residue:

1. If f(z) has a simple pole at z = a, then

2. If f(z) has a pole of order n at z = a, then

Application:
Given (-1 - j), (3 - j), (3 + j) and (-1 + j) are the vertices of a rectangle C in the complex plane

f(z) from the given data is,

 

Poleas of f(z) is

z = 0 of order n = 2, lies in side the closed curve.

z = 4 of order n = 1, lies outside the closed curve.

Pole – a point on which functional value is infinite.
If z = a is a simple pole of f(z) then the residue of f(z) at z = a is given by,

Calculation:

Given:

We have to find residue at z = 2.

Value of f(z) is infinite at x = 2. hence z = 2 is a simple pole.
Residue of f(z) at z = 2,