List of top Complex Analysis Questions on Analytic functions asked in GATE MA

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,