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The value of the integral \(\oint \frac{6z}{2z^4 - 3z^3 + 7z^2 - 3z + 5}\,dz\) evaluated over a counter-clockwise circular contour in the complex plane enclosing only the pole \(z=i\), where \(i\) is the imaginary unit, is

1. (−1 + i) π
2. (1 + i) π
3. 2(1 − i) π
4. (2 + i) π

Correct answer: (1 + i) π

Solution

Since the contour encloses only the pole at z = i, the integral equals 2\pi i times the residue at z = i. Evaluating the residue gives \((1+i)/2\), so the integral becomes \((1+i)\pi\).

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