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Let γ be the positively oriented circle in the complex plane given by {z ∈  ∶ |z - 1| = 1/2}. The line integral

 equals

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

Answer (Detailed Solution Below)

Option 1 : iπe
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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:

Singular points are given by 

z2 - 1 = 0 ⇒ z = 1, z = -1

The only singular point that lies inside γ is z = 1.

Let f(z) = 

Hence using Cauchy's Integral test

 = 2πi f(1) = 2πi × (e/2) = iπe

 Option (1) is correct. 

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