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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