List of top Complex Analysis Questions

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?

Let \( R = \{ z = x + iy \in \mathbb{C} : 0 < x < 1 \text{ and } - 11 \pi < y < 11 \pi \} \) and \( r \) be the positively oriented boundary of \( R \). Then the value of the integral \[ \frac{1}{2 \pi i} \int_r \frac{e^z}{e^z - 2} \, dz \] \text{is \(\underline{\hspace{1cm}}\) .}

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 \(\underline{\hspace{2cm}}\) .

Let \( T(z) = \frac{az + b}{cz + d}, ad - bc \neq 0 \), be the Möbius transformation which maps the points \( z_1 = 0, z_2 = -i, z_3 = \infty \) in the z-plane onto the points \( w_1 = 10, w_2 = 5 - 5i, w_3 = 5 + 5i \) in the w-plane, respectively. Then the image of the set \( S = \{ z \in \mathbb{C} : \text{Re}(z) < 0 \ \) under the map \( w = T(z) \) is}

Let \( f(z) = u(x, y) + i v(x, y) \) for \( z = x + i y \in \mathbb{C} \), where \( x \) and \( y \) are real numbers, be a non-constant analytic function on the complex plane \( \mathbb{C} \). Let \( u_x, v_x \) and \( u_y, v_y \) denote the first order partial derivatives of \( u(x, y) = \text{Re(f(z)) \) and \( v(x, y) = \text{Im}(f(z)) \) with respect to real variables \( x \) and \( y \), respectively. Consider the following two functions defined on \( \mathbb{C} \):}
\[ g_1(z) = u_x(x, y) - i u_y(x, y) \text{for} z = x + i y \in \mathbb{C}, g_2(z) = v_x(x, y) + i v_y(x, y) \text{for} z = x + i y \in \mathbb{C}. \] Then,