Let z be a complex variable. If f(z)=( z)z2(z-2) and C is the circle in the complex plane with z =3 then Cf(z)\,dz is .

GATE 2024 EC – Question 43

Engineering Mathematics · Complex Analysis · 2 marks · Multiple choice

Let $z$ be a complex variable. If $f(z)=\dfrac{\sin(\pi z)}{z^2(z-2)}$ and $C$ is the circle in the complex plane with $|z|=3$ then $\oint_Cf(z)\,dz$ is ______.

$\pi^2j$
$j\pi\left(\frac12-\pi\right)$
$j\pi\left(\frac12+\pi\right)$
$-\pi^2j$

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Correct answer: (D) $-\pi^2j$

Explanation

Both $z=0$ and $z=2$ lie inside $|z|=3$. At $z=2$, $\sin(\pi z)=\sin(\pi(z-2))\approx\pi(z-2)$, which cancels the factor $(z-2)$, so it is a removable singularity with zero residue. At $z=0$, $\sin(\pi z)\approx\pi z$, so it is a simple pole with residue $\lim_{z\to0}\dfrac{\sin\pi z}{z(z-2)}=\dfrac{\pi}{-2}$. The integral is $2\pi j\left(-\dfrac\pi2\right)=-\pi^2j$.