The value of $\oint_{C} \frac{s i n z}{z} d z,$ where the contour of the integration is a simple closed curve around the origin is

0

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2 π j

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α

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\frac{1}{2 π j}

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Solution

The correct option is A $0$
$z = 0$ is a simple pole of integrand and $f (z) = \frac{s i n z}{z}$
So Res $f (z); (z = 0) = {lim}_{Z \to 0} [z f (z)] = {lim}_{Z \to 0} [s i n (z)] = 0$
$∵ Z = 0$ lies with in the contour. Hence by Cauchy Residue theorem.
$\int_{C} \frac{s i n z}{Z} d z = 2 π i$ [Residue]
$= 2 π i [0]$
$= 0$

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