Exercise 4.3 (Singularities residues residue theorem)

EXERCISE 4.3.

SINGULARITIES - RESIDUES - RESIDUE THEOREM

I. Zeros, poles (or) singularities

1. Find the zeros of ƒ (z) = sin z - z / z³

Ans. no zeros

2. Find the poles of f (z): z² + 1 / 1 - z²

Ans. -1, 1

3. Find the poles of f (z) = cosec z

Ans. ± nπ, n = 0, 1, 2, ...

4. Find the poles of f (z) = cot z

Ans. ± nπ, n = 0, 1, 2, ...

5. Find the poles of f (z) = tanz

Ans. ± (2n+1), π/2,  n = 0, 1, 2, ...

6. Find the poles of f (z) = 1 / sin (1 / z - ɑ)

Ans. z = ɑ + 1 / n π. n = ±1, ± 2, ...

7. Find the poles of f (z) = z³ – 1 / z³ + 1

Ans. – 1, 1/2 ± i √3/2

8. Find the poles of f (z) = z / cos z

Ans. ±(2n + 1) π/2, n = 0,1,2,…

9. Find the nature of the singularities of (a) z / sin z (b) cos z / z

Ans. (a) z = 0 is a removable singularity, z = n π, n = 1, 2, ... are poles

(b) z = 0 is a simple pole.

10. Find the orders of the pole z = 0 of the following functions:

(i) e^z / z

Ans. z = 0 is a simple pole

(i) e^z / z²

Ans. z = 0 is a pole of order 2

 (iii) 1- sin z / z³

Ans. z = 0 is a pole of order 3.

11. Find the nature of the singularities of z²/ (z - 1)² (z - 2)

Ans. z = 1 is a double pole and z = 2 is a simple pole

12. Find the nature of the singularity of f (z) = (z − 3) sin (1/ z + 2)

 Ans. z = -2 is an essential singularity

13. Classify the singularity of the function f (z) = z – 1 / z² sin (1/z – 3)

Ans. z = 1 is a simple zero and z = 0 is a pole of order 2

14. Find the poles of f (z) = cot π z / (z – 1)³

Ans. z = 1 / n π + 1, n = 0,1,2, … and z = ɑ is a pole of order 3

15. Find the poles of f (z) = cosec z / (z - 1)³

Ans. z = n π, n = 0, ±1, ±2, ... and z = 1 is a pole of order 3

II. Residue

1. Calculate the residue of z + 1 / z² – 2z at its poles.

Ans : -1 / 2, 3 / 2

2. Find the residue of f (z): e^z / z² (z² + ɑ)

Ans. 1/9 – [sin 3 + i cos 3 / 54]

3. Find the residue of 1 / (z²  + ɑ²) ²  at z = ɑ i

Ans. –i / 4 ɑ³

4. Find the residue of ze^z /  (z - 1)³ at its poles.

Ans. 3e / 2

5. Find the residue of f (z) = z² / z² ɑ² at its singularities.

Ans. - iɑ / 2 ,  iɑ / 2

6. Find the residues of f (z) = z⁴ / (z – 1)⁴ ( z – 2) ( z – 3) at its singularities.

Ans : -16, 81/16, 175/16

7. Find the residue of f (z) = 1 / ( 1 + z²)⁴ at z = i

Ans. – 5i / 32

8. Find the residue of f (z) = z sin z / (z – π)³ at its singularity.

Ans. residue at z = π is -1

9. Determine the poles and residues at each of the function

f(z) = z² /  (z − 1)³ (z + 2)

Ans : 4/9 , 4/27

10. Find the residue of f (z) = z² sin (1/z)  at z = 0

Ans. -1 / 6

11. Calculate the residues of f (z) = z³ - 2z / (z + 1)² (z² + 4)

Ans. 7 / 25 , -3 / -3 + 4 4i , 3 / 3 + 4i

12. Determine the poles of the following functions and the residue at each pole.

III. Cauchy's residue theorem

1. Determine the poles of the function f(z) = z2 / (z – 1)2 (z + 2) and residue at each pole. Hence, evaluate where C is the  circle |z| = 2.5.

Ans : 2πi

2. Evaluate  where C is the  circle

3. Evaluate the following integral

4. Obtain Laurent's expansion for the function f(z) = 1 / z² sinh z and evaluate  where C is the circle |z - 1| = 2

5. If C is the circle |z|= 2 evaluate, using residue theorem

Ans : 10 πi

6. If C is the circle |z - i|= 2 evaluate 

Ans. – πi / 3

7. Evaluate  where C is |z - i| = 3.

Ans. -1

8. If C is ❘z – 1 | = 1, evaluate 

Ans : -4 πi / 9

Evaluate the following: