Question
The value of the integral \(\mathop \oint \limits_c \frac{{2z + 5}}{{\left( {z - \frac{1}{2}} \right)\left( {{z^2} - 4z + 5} \right)}}dz\) over the contour \(\left| z \right| = 1\), taken in the anti – clockwise direction, would be
The correct answer is
\(\frac{{48\pi i}}{{13}}\)
Only z=1/2 lies inside the unit circle. Its residue is \(6/[(1/2)^2-4(1/2)+5]=24/13\). The integral is2πi times this residue, or 48πi/13.
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