If $C$ denotes the counterclockwise unit circle, the value of the contour integral $\frac{1}{2 π j} \oint_{C} R e z d z$ is

0.5

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Solution

The correct option is A 0.5
$C : | z | = 1 \Rightarrow z = e^{i θ}$ & $d z = j e^{j θ} d θ$ where $0 \leq θ \leq 2 π$
$\frac{1}{2 π j} \oint_{C} R e z d z = \frac{1}{2 π j} \int_{0}^{2 π} R e (e^{j θ}) j e^{j θ} d θ$
$= \frac{1}{2 π j} \int_{0}^{2 π} c o s θ . j (c o s θ + j s i n θ) d θ$
$= \frac{1}{2 π} [\int_{0}^{2 π} c o s^{2} θ d θ - \int_{0}^{2 π} c o s θ s i n θ d θ]$
$= \frac{1}{2 π} (π - 0) = \frac{1}{2}$

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