Problems based on cauchy's integral theorem

I. (a) PROBLEMS BASED ON CAUCHY'S INTEGRAL THEOREM

 

Example 4.1.a.1. State Cauchy's integral theorem. [Anna, May 1998] [A.U A/M 2015 R13, R8] [A.U April 2017 R-15 U.D]

Statement: If a function f(z) is analytic and its derivative f ' (z) is continuous at all points inside and on a simple closed curve C, then 

Example:  since sinz is an analytic function in C. [A.U A/M 2019 (R17)]

 

Example 4.1.a.2. Evaluate  where C is the circle | z | = 1.

[A.U M/J 2014, N/D 2014]

Solution : Let f (z) = z/z - 2

Here, z= 2 lies outside C.

f(z) is analytic inside and on C.

f '(z) is continuous inside C.

Hence, by Cauchy's integral theorem 

 

Example 4.1.a.3. Evaluate  if C is the circle | z | = 1

Solution : Given: 

Let f (z) = 1 / z – 3/2

Here, z = 3/2 lies outside C

f(z) is analytic inside and on C.

f '(z) is continuous inside C

Hence, by Cauchy's integral theorem 

 

Example 4.1.a.4. Evaluate  where C is | z | = ½ [A.U A/M 2018 R-17]

Solution: Let f (z) = 3z² + 7z + 1/z – (-1)

Here, z= -1 lies outside C

Complex Integration enpre

f(z) is analytic inside and on C.

f' (z) is continuous inside C.

Hence, by Cauchy's integral theorem 

 

Example 4.1.a.5. Evaluate  where C is | z-2 |= 4.

Solution: Let f (z) = (2-2)ⁿ

f(z) is analytic inside and on C.

 By Cauchy's integral theorem, we get 

 

Example 4.1.a.6. Evaluate where C is | z | = 1. [Anna, Nov. 2001]

Solution: Let f (z)  = e^l/z

f(z) is analytic inside and on C.

Hence, by Cauchy's integral theorem we get 

 

Example 4.1.a.7. Can the Cauchy-integral theorem be applied for evaluating the following integrals? Hence evaluate these integrals. 

 [A.U M/J 2016 R-8]

Solution: (1) 

Let f (z) = z²

f(z) is analytic inside and on C. f' (z) is continuous inside C. Hence, by Cauchy's integral theorem 

(2) 

Let f (z) = tanz = sin z / cos z is analytic except at z = ± π/2 , ±3π/2, ...

All these points lie outside C. f'(z) is continuous inside C.

Hence, by Cauchy's integral theorem 

(3) 

Let f (z) = e^sinz2, f (z) is analytic inside and on C.

f' (z) is continuous inside C.

Hence, by Cauchy's integral theorem 

 

Example 4.1.a.8. Evaluate , where C is | z| = 1/2

Solution:

Given: 

z² + 1 = 0

2²  = -1

z = ±i both lies outside C

 by Cauchy's Integral theorem.

 

Example 4.1.a.9. Evaluate where C is | z + 3 | = 1 [A.U A/M 2017 R-08]

Solution: Given: 

Here, C is a circle with centre (-3, 0) radius 1.

Let f (z) = e^z /z - 1

Here, z= 1 lies outside C.

f (z) is analytic inside and on C. f' (z) is continuous inside C.

Hence, by Cauchy's integral theorem 

 

Example 4.1.a.10. Evaluate  where C is the unit circle with centre as origin. [A.U A/M 2017 R-13]

Solution: Given: 

Here, C is the unit circle with centre as origin.

Let f (z) = e^z/z - 2

Here, z = 2 lies outside C

f (z) is analytic inside and on C. f' (z) is continuous inside C.

Hence, by Cauchy's integral theorem 

 

Example 4.1.a.11 Evaluate  where C is the unit circle | z |= 1 [A.U April 2016 R-15 U.D]

Solution: Given: 

Here, C is the unit circle with centre as origin.

Let f (z) = e^z/z - 3

Here, z = 3 lies outside C

f(z) is analytic inside and on C. f' (z) is continuous inside C.

Hence, by Cauchy's integral theorem