Question

The value of the contour integral ${\oint }_{|z-j|=2}\frac{1}{z^2+4}dz$ in positive sense is

• A. $j\pi /2$
• B. $-\pi /2$
• C. $-j\pi /2$
• D. $\pi /2$

Answer 1

The correct option is D $\pi /2$

${\oint }_{|z-j|=2}\frac{1}{z^2+4}dz={\oint }_{|z-j|=2}\frac{dz}{(z-2j)(z+2j)}$
Pole are, $z^2+4=0\Rightarrow z=\pm 2j$
$z=-2jliesoutsidethecurve|z-j|=2$
$\therefore {\oint }_{|z-j|=2}\frac{1}{z^2+4}dz=2\pi j\left(Sumofresidues\right)$
$=2\pi j\left(\frac{1}{4j}\right)=\frac{\pi }{2}$