GATE 2026 IN – Question 13
Consider a function $f(z)=z^2+z+1$, where $z\in\mathbb{C}$ is a complex variable. A simple closed contour $\gamma$ in the $z$-plane encloses the point $z=1+0j$. The value of $\displaystyle\oint_\gamma\frac{f(z)}{z-1}\,dz$ is ________.
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Correct answer: (A) $6\pi j$
Explanation
The numerator is a polynomial, hence analytic inside and on the contour. For the usual counterclockwise orientation, Cauchy’s integral formula gives $\oint_\gamma f(z)/(z-1)\,dz=2\pi j f(1)$.
Here $f(1)=1+1+1=3$, so the integral is $6\pi j$, **A**. The printed question does not explicitly specify orientation; clockwise traversal reverses the sign.