Test: Cauchy’s Integral Theorem - 1 - Civil Engineering (CE) MCQ

Given that The singular point is at z = 2.

If z = 2 is lies inside of the contour then the value of f(z) = Residue of f(z) at z = 2
∴ 
= e² = 7.39
If z = 2 is lies outside of the contour, it means that the given function is analytic.
Then the value of f(z) = 0

Given that,

C : |Z – (-2)| = 1
⇒ C : |Z + 2| = 1

 

poles of f(Z) are z² – 4 = 0

⇒ Z = ±2

Z = 2 is lies outside the curve C.

f(Z) = 2πi [residue at Z = -2]

Concept:
If f(z) is analytic within & on a closed curve C and “a” is a point inside the curve C then,
 we consider f(z)/(z-a) to be analytic at all points within C except at z=a.

If multiple singularities occur then we use this;

Calculation:
We have to find this 
Comparing the above with equation ..(1), we conclude that f(z) =1.

If we take differentiation then f’(a) = 0.

This implies that

Hence option 2 is correct 

Cauchy's Theorem:

If f(z) is single-valued and an analytic function of z and f'(z) is continuous at each point within and on the closed curve c, then according to the theorem, 
Cauchy's Integral Formula:
For Simple Pole:
If f(z) is analytic within and on a closed curve c and if a (simple pole) is any point within c, then

For Multiple Pole:

If f(z) is analytic within and on a closed curve c, and if a (multiple poles) are points within c, then

Concept:

For a given complex function with poles, the complex integralis given by

Residue theorem as;
C⁡f(z)dz = 2π i × {Sum of residue of poles in side or onC}

Calculation:

Contour: |z|= 3
Simple pole, z = 2 and it is lies inside the contour.
Residue of f(z) at z = 2 is,
= 2³ − 2(2) + 3 = 7
f(z) = 2πi(7) = 14πi

(z - 1)³ (z - 3) = 0

⇒ z = 1, z = 3

The function has a simple pole at z = 3 and has a multiple pole at z = 1

Both z = 1, and z = 3 are inside the region C

According to Cauchy’s Residue theorem
⁡ f (z)dz = 2πi [sum of the residues at the poles in side ′C′]
If z = a is a pole of order ‘m’, then residue of f(z) at z = a is,

Concept:
Cauchy Integral Theorem:

 

Where z = a be any point inside the close region.

Cauchy’s Residue Theorem:
= Sum of Residue at Pole or singularity with in the region

Res at z = a

Calculation:
We know that e^ix = cos x + i sin⁡ x
Let x replace by z.

Now, z² + 2z + 2 = 0 then roots of z are:


 

z = -1 ± i is the only pole lying in f(z) > 0

Here n = 1

Concept:

Cauchy's Integral Formula:

For Simple Pole:

If f(z) is analytic within and on a closed curve c and if a (simple pole) is any point within c, then

Calculation:
Given:
where C represents unit circle i.e. radius is unity.

The above equation can be written in standard form i.e. 
Therefore f(z) = and a = 0.

The pole of the given function is at z = 0, and lie inside the circle.

Cauchy's Integral Formula:

All the statements given are verified by different theorems, 

Theorem: If a sequence converges then all subsequences converge and all convergent subsequences converge to the same limit (Option 3)

Theorem: Every bounded sequence has a convergent subsequence (Option 4)

Theorem: If {a_n}_n∈N is a sequence that either has a subsequence that diverges or two convergent subsequences with different limits then {a_n}_n∈N is divergent (Option 1)

Theorem: 

1) A sequence {a_n} of real numbers is called a Cauchy sequence if for each ϵ > 0 there is a number N ∈ N so that if m, n > N then |a_n − a_m| < ϵ.

2) If a real sequence {an} converges, then for every ε > 0, there exists N ∈ N such that |a_n − a_m| < ε ∀ n,m ≥ N

3) Convergent sequences are Cauchy sequences.

A Cauchy sequence of real numbers is bounded (Option 2 is false)

A sequence is a convergent sequence if and only if it is a Cauchy sequence.

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 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

Application:

The simple poles are: z = 0, 2

The given region is a unit circle.

The residue at z = 2 is zero as it lies outside the given region.

The reside at z = 0, is given by

The value of the given integral =