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Question

The value of r3z5(z1)(z2)dz along a closed path Γ is equal ot 4πi, where z=x+iy and i=1.
The correct Γ is

A
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B
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C
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D
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Solution

The correct option is B
Method I:
Let f(z)=3z5(z1)(z2) so poles are z=1 & 2
R1=Res f(z)(z=1)=limz1((z1)f(z))=2R1=Res f(z)(z=2)=limz2((z2)f(z))=1Let us assume that z=1 lies inside given contour and z= lies outisde it.
Then by C - R - T,
f(z)dz=2πi (sum of residues at poles inside)
=2πi(2)=4πi
Hence verified,
So our assumption that z=1 lies inside Γ and z=2 lies outside Γ so correct path of Γ is defined by (b)

Method II:
Using Cauchy's formula,
3z5(z1)(z2)dz=3z5z2z1dz
=j2π[3z5z2]atz=1
=j2π[3512]
=j2π[21]=j4π
Hence, z=1 lies inside the closed path Γ. and z=2 lies outside the closed path Γ.

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