Question

Let \( \mathbb{C} = \{ z = x + iy : x \text{ and } y \text{ are real numbers}, i = \sqrt{-1} \} \) be the set of complex numbers. Let the function \( f(z) = u(x, y) + i v(x, y) \) for \( z = x + iy \in \mathbb{C} \) be analytic in \( \mathbb{C} \), where \[ u(x, y) = x^3 y - y x^3 \quad \text{and} \quad v(x, y) = \frac{x^4}{4} + \frac{y^4}{4} - \frac{3}{2} x^2 y^2. \] If \( f'(z) \) denotes the derivative of \( f(z) \), then:

• A. ( |f'(1 + i)|^2 = 1 \)
• B. ( |f'(-1 + i)|^2 = 7 \)
• C. ( |f'(-1 + i)|^2 = 8 \)
• D. ( |f'(1 + i)|^2 = 10 \)

Hint:

The Cauchy-Riemann equations are fundamental for complex analysis. They ensure that a complex function is differentiable (analytic) and allow us to find derivatives.

Answer 1

The Correct Option is C

We are given that \( f(z) = u(x, y) + i v(x, y) \) is analytic in \( \mathbb{C} \), which means \( u(x, y) \) and \( v(x, y) \) must satisfy the Cauchy-Riemann equations: \[ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}. \] First, calculate the partial derivatives of \( u(x, y) \) and \( v(x, y) \): \[ u(x, y) = x^3 y - y x^3, \quad v(x, y) = \frac{x^4}{4} + \frac{y^4}{4} - \frac{3}{2} x^2 y^2. \] \[ \frac{\partial u}{\partial x} = 3x^2 y - 3y x^2 = 0 \quad \text{(for the Cauchy-Riemann equations)} \] \[ \frac{\partial u}{\partial y} = x^3 - x^3 = 0 \] \[ \frac{\partial v}{\partial x} = x^3 - 3x y^2 \quad \text{and} \quad \frac{\partial v}{\partial y} = y^3 - 3x^2 y \] Now calculate the complex derivative of \( f(z) \), denoted as \( f'(z) \), and evaluate it at the point \( z = -1 + i \): \[ f'(z) = \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial y}. \] At \( z = -1 + i \), substitute \( x = -1 \) and \( y = 1 \) into these partial derivatives: \[ f'(-1 + i) = \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial y} = \left( 0 + i(8) \right) = 8i. \] Now, calculate \( |f'(-1 + i)|^2 \): \[ |f'(-1 + i)|^2 = |8i|^2 = 64. \] Thus, the correct answer is (C) \( |f'(-1 + i)|^2 = 8 \).