Question

Which of the following statements are correct:
A. $e^{-x}(x \sin y - y \cos y)$ is harmonic B. $f(z) = u + iv$ is analytic, where $u = x, v = y$.
C. $f(z) = z^{1/2}$ is analytic on $\mathbb{C}$. D. $f(z) = \frac{1}{z}$ is not analytic on $\mathbb{C}$. E. $f(z) = \sin(z^3)$ is bounded.
Choose the correct answer from the options given below:

• A. A, C, E Only
• B. A, B, D Only
• C. A, B, C Only
• D. A, D, E Only

Hint:

Remember Liouville's Theorem: Non-constant entire functions like $\sin(z^3)$ or $e^z$ are NEVER bounded on $\mathbb{C}$!

Answer 1

The Correct Option is B

Step 1 : Concept:
This question tests basic properties of complex analytic functions, harmonic functions, domain of analyticity, and Liouville's theorem regarding bounded entire functions.

Step 2 : Key Formulas and Approach:
1. A function $u(x,y)$ is harmonic if it satisfies Laplace's equation: \[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \] 2. Cauchy-Riemann equations for analyticity of $f(z) = u + iv$: \[ u_x = v_y \quad \text{and} \quad u_y = -v_x \] 3. A function is analytic on $\mathbb{C}$ if it is differentiable at every point in $\mathbb{C}$.
4. Liouville's Theorem: Any entire function that is bounded on $\mathbb{C}$ must be constant.

Step 3 : Step-by-step Explanation:

• Statement A:
Let $u(x,y) = e^{-x}(x \sin y - y \cos y)$.
First partial derivatives: \[ u_x = -e^{-x}(x \sin y - y \cos y) + e^{-x} \sin y = e^{-x} (\sin y - x \sin y + y \cos y) \] \[ u_y = e^{-x}(x \cos y - \cos y + y \sin y) \] Second partial derivatives: \[ u_{xx} = -e^{-x}(\sin y - x \sin y + y \cos y) + e^{-x}(-\sin y) = e^{-x} (-2 \sin y + x \sin y - y \cos y) \] \[ u_{yy} = e^{-x}(-x \sin y + \sin y + \sin y + y \cos y) = e^{-x} (2 \sin y - x \sin y + y \cos y) \] Adding $u_{xx}$ and $u_{yy}$: \[ u_{xx} + u_{yy} = e^{-x} [(-2 \sin y + x \sin y - y \cos y) + (2 \sin y - x \sin y + y \cos y)] = 0 \] Since $u_{xx} + u_{yy} = 0$, $u(x,y)$ is harmonic. Statement A is correct.

• Statement B:
Here $u = x$ and $v = y$, so $f(z) = x + iy = z$.
Check Cauchy-Riemann equations: $u_x = 1, v_y = 1 \implies u_x = v_y$ and $u_y = 0, v_x = 0 \implies u_y = -v_x$.
Since $f(z) = z$ is differentiable everywhere on $\mathbb{C}$, it is analytic. Statement B is correct.

• Statement C:
$f(z) = z^{1/2}$ is a multi-valued function with a branch point at $z = 0$. It is not single-valued or differentiable on the entire complex plane $\mathbb{C}$. Statement C is incorrect.

• Statement D:
$f(z) = \frac{1}{z}$ is not defined or differentiable at $z = 0$, so it is not analytic on the entire complex plane $\mathbb{C}$. Statement D is correct.

• Statement E:
$f(z) = \sin(z^3)$ is an entire non-constant function. By Liouville's theorem, it cannot be bounded. For instance, along $z = i y$, $|\sin(-i y^3)| = |\sinh(y^3)| \to \infty$ as $y \to \infty$. Statement E is incorrect.

Step 4 : Final Answer:
Statements A, B, and D are correct. Therefore, option (B) is the correct answer.