Consider the integralWhere C is a counter-clockwise oriented circle defined as |x - i| = 2. The value of the integral isa)b)c)d)Correct answer is option 'B'. Can you explain this answer?

Question

Answer

Concept:
Residue Theorem:
If f(z) is analytic inside and on a closed curve ‘C’ except at a finite number of singularities inside ‘C’

Then:

∮f(z) dz = 2πi (sum of residues)

Singularity: A point where the function f(z) fails to be analytic

Analysis:

x = 0 pole of order 2
x = 2i, -2i
Given curve or region is |x - i| = 2

x = -2i lies outside |x - i| = 2
So the singular points are x = 0 & x = 2i
Residue at x = 0

Residue at x = 2i

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