List of top Complex Analysis Questions 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)

Let \(f(z) = \dfrac{z}{1-z}\) and \(g(z) = \dfrac{1+z}{1-z}\) be two Mobius transformations defined on the unit disc \(D = \{z \in \mathbb{C} : |z| < 1\}\). Consider the following statements:
\(S_1\): \(f(D) \subseteq g(D)\)
\(S_2\): \(g(D) \subseteq f(D)\)
Which of the following statements is/are CORRECT?

The value of the contour integral \[ \int_{\Gamma} \frac{(1-z)(3\cos z+5\sin z)}{1-z^{101}}\,dz, \] where \(\Gamma = \left\{z\in\mathbb{C} : |z|=\frac{7}{9}\right\}\), oriented in the counter-clockwise direction, is equal to

Let \(f(z) = \displaystyle\sum_{n=1}^{\infty} 5^{-n}\cos(nz)\). On which one of the following domains, does \(f(z)\) represent an analytic function?

For \(a_n \in \mathbb{C}\), \(n = 0,1,2,\ldots\), the power series \[ \sum_{n=0}^{\infty} a_n (z-2)^n \] converges at \(z = 5\) and diverges at \(z = -1\). Then the radius of convergence of this power series is equal to ______. (Answer in integer)

Let \( f: \mathbb{C} \to \mathbb{C} \) be defined by \( f(z) = |z|^2 - 5\bar{z} + 2 \).
Which one of the following is TRUE?

Let \( D = \{ z \in \mathbb{C} \mid |z| < 1 \} \) denote the unit disc in the complex plane \( \mathbb{C} \).
Let \( f: D \to D \) be an analytic function which satisfies \( f(0) = 0 \). Then which one of the following is a possible value of \( f'(0) \)?