Question

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,

• A. both \( g_1(z) \) and \( g_2(z) \) are analytic in \( \mathbb{C} \)
• B. \( g_1(z) \) is analytic in \( \mathbb{C} \) and \( g_2(z) \) is NOT analytic in \( \mathbb{C} \)
• C. \( g_1(z) \) is NOT analytic in \( \mathbb{C} \) and \( g_2(z) \) is analytic in \( \mathbb{C} \)
• D. neither \( g_1(z) \) nor \( g_2(z) \) is analytic in \( \mathbb{C} \)

Hint:

For a function to be analytic, the Cauchy-Riemann equations must hold. Functions of the form \( g_1(z) \) derived from the partial derivatives of \( u(x, y) \) and \( v(x, y) \) may be analytic, while others may not.

Answer 1

The Correct Option is B

We are given that \( f(z) = u(x, y) + i v(x, y) \) is analytic in \( \mathbb{C} \), and we need to determine the analyticity of the functions \( g_1(z) \) and \( g_2(z) \). To check analyticity, we apply the Cauchy-Riemann equations. The Cauchy-Riemann equations state that for a function \( f(z) = u(x, y) + i v(x, y) \) to be analytic, the following must hold: \[ u_x = v_y \text{and} u_y = -v_x. \] 1. For \( g_1(z) = u_x(x, y) - i u_y(x, y) \), the Cauchy-Riemann equations are satisfied, as it is derived directly from the partial derivatives of \( u(x, y) \) and \( v(x, y) \), ensuring that \( g_1(z) \) is analytic in \( \mathbb{C} \). 2. For \( g_2(z) = v_x(x, y) + i v_y(x, y) \), the Cauchy-Riemann equations are not satisfied, which implies that \( g_2(z) \) is not analytic in \( \mathbb{C} \). Thus, the correct answer is (B).