Discussion
Value of \(\oint_{c} \frac{dz}{(z+2)}\), \(c : |z| = 1\) is :
Answer: B
The integral is zero.
The integrand \(1/(z+2)\) has a single simple pole, at \(z = -2\). The contour is the unit circle \(|z| = 1\), and since \(|-2| = 2 \gt 1\) the pole lies outside it. Inside and on the contour the function is analytic, so by Cauchy's theorem
\[ \oint_{|z|=1} \frac{dz}{z+2} = 0 \]
The value \(2\pi i\) would be the answer only if the pole were enclosed (residue 1 times \(2\pi i\)); \(-2\pi i\) is that with the contour traversed clockwise, and \(\pi i\) arises for a pole lying on a contour, in the principal-value sense. None of these applies here.
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