Test: Residue Theorem - 1 - Civil Engineering (CE) MCQ

Simple poles, z = ±1
According to Cauchy’s residue theorem,
⁡f(z)dz = 2πi [sum of residues]

At, z = +1, residue is

At, z = -1 residue is zero as z = -1 lies outside the curve C.

∮1/z² dz
By Cauchy’s residue theorem,
∮ f(z)dz = 2πi [sum of residues]
f(z) = 1/z²

Poles are, Z = 0

Residue at Z = a, is given by

Where n is the order pole.

Here, z = 0 is second order pole.

Residue at z = 0,

Residue of


z = x + iy and = x − iy i.e.u = x and v = − y
u_x = 1 and v_y = −1
u_X ≠ v_y → is not Analytic Analytic

Concept:
For simple poles at z = a, b, c…
Residue of 

For multiple poles at z = a, a, a … n times
{Residue of 
Calculation:
Given, f (z) = 
For a simple pole at z = 4
Residue of 

For multiple pole (n = 3) at z = -1
Residue will be

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

Important Points:

Cauchy’s Theorem:

If f(z) is an analytic function and f’(z) is continuous at each point within and on a closed curve C, then

Cauchy’s Integral Formula:
If f(z) is an analytic function within a closed curve and if a is any point within C, then

Poles of f(z) are z = 1, 2

Both Z = 1 and 2 are inside the curve |z| = 4

= 2πi [sum of residues at z = 1 and z = 2]

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

Residue at z = 1,

Residue at z = 2,
 
 

Concept:

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:
Given:

Poles are simple and located at z = 0, z = 1, and z = 2

At z = 0, the residue is:

At z = 1, the residue is:

At z = 2, the residue is:

Concept:

Cauchy’s Theorem:

If f(z) is an analytic function and f’(z) is continuous at each point within and on a closed curve C, then

Cauchy’s Integral Formula:

If f(z) is an analytic function within a closed curve and if a is any point within C, then

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

Application:
Given function is 

Poles: z = 1, -1
|z – 1| = 1
⇒ |x – 1 + iy| = 1

The given region is a circle with the centre at (1, 0) and the radius is 1.

Only pole z = 1, lies within the given region.

Residue at z = 1 is,  
The value of the integral = 2πi × 0.5 = πi

Concept:

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, 

Here we have n = 2 and a = 2