List of top Complex Analysis Questions asked in GATE MA

The value of the integral \[ \int_{C} \frac{z^{100}}{z^{101} + 1} \, dz \] where C is the circle of radius 2 centered at the origin taken in the anti-clockwise direction is

Consider the real function of two real variables given by
\[ u(x, y) = e^{2x}[\sin 3x \cos 2y \cosh 3y - \cos 3x \sin 2y \sinh 3y]. \] Let \( v(x, y) \) be the harmonic conjugate of \( u(x, y) \) such that \( v(0, 0) = 2 \). Let \( z = x + iy \) and \( f(z) = u(x, y) + iv(x, y) \), then the value of \( 4 + 2i f(i\pi) \) is

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 radius of convergence of the series \[ \sum_{n \geq 0} 3^{n+1} z^{2n}, \quad z \in \mathbb{C} \] is __________ (round off to TWO decimal places).

Let \( T \) be a Möbius transformation such that \( T(0) = \alpha, T(\alpha) = 0 \) and \( T(\infty) = -\alpha \), where \( \alpha = \frac{-1 + i}{\sqrt{2}} \). Let \( L \) denote the straight line passing through the origin with slope \( -1 \), and let \( C \) denote the circle of unit radius centered at the origin. Then, which of the following statements are TRUE?