Discussion
Evaluate \( \displaystyle\oint_{|z|=1} \frac{dz}{z} \).
Answer: A
Put \( z = e^{i\theta} \), so \( dz = ie^{i\theta}d\theta \):
\[ \oint_{|z|=1}\frac{dz}{z} \;=\; \int_{0}^{2\pi} \frac{ie^{i\theta}}{e^{i\theta}}\,d\theta \;=\; i\int_{0}^{2\pi} d\theta \;=\; 2\pi i. \]
It is non-zero because the origin, where the integrand fails to be analytic, lies inside the contour
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