Gate 2016 EE (Set 1) Q5

 

5. The value of the integral  \int(2z+5)/(z-1/2)(z^2-4z+5) dz over the contour |z| = 1

, taken anti-clockwise direction, would be:--------

Solution:

Singular points, equating the denominator to 0

 (z-1/2)(z^2-4z+5)=0

Singular points are, z= 1/2 and z=2\pm i

Using Cauchy’s integrated formula,

 (1/2\pi)^(-1)* \int (2z+5)/((z^2-4z+5) ) dz |z=1/2

 2\pi * [2(1/2)+5]/((1/2)^2-4(1/2)+5)=48\pi /13

 

 

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