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

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

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## 10 Questions MCQ Test Engineering Mathematics - Test: Residue Theorem - 1

Test: Residue Theorem - 1 for Civil Engineering (CE) 2024 is part of Engineering Mathematics preparation. The Test: Residue Theorem - 1 questions and answers have been prepared according to the Civil Engineering (CE) exam syllabus.The Test: Residue Theorem - 1 MCQs are made for Civil Engineering (CE) 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Residue Theorem - 1 below.
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Test: Residue Theorem - 1 - Question 1

### The value of the integral where z is a complex number and C is a unit circle with centre at 1 + 0j in the complex plane is __________ .

Detailed Solution for Test: Residue Theorem - 1 - Question 1

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.

Test: Residue Theorem - 1 - Question 2

### The value of ∮1/z2 dz, where the contour is the unit circle traversed clockwise, is

Detailed Solution for Test: Residue Theorem - 1 - Question 2

∮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,

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Test: Residue Theorem - 1 - Question 3

### Let z = x + iy be a complex variable. Consider that contour integration is performed along the unit circle in anticlockwise direction. Which one of the following statements is Not True?

Detailed Solution for Test: Residue Theorem - 1 - Question 3

Residue of

z = x + iy and = x − iy i.e.u = x and v = − y
ux = 1 and vy = −1
uX ≠ vyis not Analytic Analytic

Test: Residue Theorem - 1 - Question 4

The residues of a function f(z) = are:

Detailed Solution for Test: Residue Theorem - 1 - Question 4

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

Test: Residue Theorem - 1 - Question 5

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

Detailed Solution for Test: Residue Theorem - 1 - Question 5

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

Test: Residue Theorem - 1 - Question 6

If C is a circle |z| = 4 and f(z) = then ∮ f (z) dz is

Detailed Solution for Test: Residue Theorem - 1 - Question 6

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,

Test: Residue Theorem - 1 - Question 7

The residues of a complex function X (z) = at its poles are

Detailed Solution for Test: Residue Theorem - 1 - Question 7

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:

Test: Residue Theorem - 1 - Question 8

Integration of the complex function f (z) =  in the counterclockwise direction, around |z – 1| = 1, is

Detailed Solution for Test: Residue Theorem - 1 - Question 8

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

Test: Residue Theorem - 1 - Question 9

Residue at z = 2 of  is

Detailed Solution for Test: Residue Theorem - 1 - Question 9

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,

Test: Residue Theorem - 1 - Question 10

The residue of the function f (z) =  at z = 2 is

Detailed Solution for Test: Residue Theorem - 1 - Question 10

Here we have n = 2 and a = 2

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