List of top Complex Analysis Questions on Rouche’s theorem asked in GATE MA

The number of zeros of the complex polynomial \( z^6 + 5z^3 + 4z + 11 \) in the annulus \( \{z \in \mathbb{C} : 1 < |z| < 3\} \) is equal to ______. (Answer in integer)

The number of zeros of the polynomial \[ 2z^7 - 7z^5 + 2z^3 - z + 1 \] in the unit disc \( \{ z \in \mathbb{C} : |z|<1 \} \) is __________.

The number of zeros (counting multiplicity) of \( P(z) = 3z^5 + 2iz^2 + 7iz + 1 \) in the annular region \( \{ z \in \mathbb{C} : 1 < |z| < 7 \} \) is \(\underline{\hspace{2cm}}\) .