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Let γ be the positively oriented circle in the complex plane given by {z ∈  ∶ |z - 1| = 1}. Then equals

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CSIR-UGC (NET) Mathematical Science: Held on (26 Nov 2020)
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  1. 3
  2. 1/3
  3. 2
  4. 1/2

Answer (Detailed Solution Below)

Option 2 : 1/3
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Detailed Solution

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Concept:

Cauchy Integral Theorem:

If a complex function f(z) is analytic within and on a closed contour C inside a simply-connected domain, and if a is any point in the middle of C, then

f(a) = 

Explanation:

 = 

So poles are given

(z - 1)(z2 + z +1) = 0 ⇒ z = 1, z =  = 

Poles inside γ is z  = 1

Hence f(z) =  is analytic inside γ

Therefore by Cauchy Integral theorem, 

 = f(1) =  = 1/3 

 Option (2) is correct 

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