GATE Electrical Engineering (EE) 2019 Solved Paper

Question : 38 of 65

Marks: +1, -0

The closed loop line integral

∮_{|z|=5} \;{z^{3}+z^{2}+8}/{z+2} d z

evaluated counter-clockwise, is

Solution:

∫↙{|z|-5} \;{z^{3}+z^{2}+8}/{z+2}

Let

f(z)=\;{z^{3}+z^{2}+8}/{z+2}

{f}( {z})

is not analytic at

{z}=-2

and

z=-2

lies in side of given curve

|z|=5

[\Res {f}( {z})]_{ {z}--2} =L t_{ {z}-2}( {z}+2) \;{ {z}^{3}+ {z}^{2}+8}/{( {z}+2)}

=-8+4+8

=4

By cauchy's residue theorem,

∮_{c} \;{z^{3}+z^{2}+8}/{z+2} d z=2 π j ×(

sum of residues of points inside of

C)

=2 π {j} × 4=8 π {j}

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