Question

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R
Assertion A : The function $f : \mathbb{C} \to \mathbb{C}$ defined by $f(z) = z - \bar{z}$ is analytic at every point in $\mathbb{C}$ Reason R : $f$ does not satisfy the Cauchy - Riemann equations at any point.
In the light of the above statements, choose the correct answer from the options given below

• A. Both A and R are true and R is the correct explanation of A
• B. Both A and R are true but R is NOT the correct explanation of A
• C. A is true but R is false
• D. A is false but R is true

Hint:

Any function containing $\bar{z}$ explicitly (such as $z - \bar{z} = 2i \text{Im}(z)$) fails C-R equations almost everywhere and is non-analytic!

Answer 1

The Correct Option is D

Step 1: Concept
A complex function $f(z) = u(x, y) + i v(x, y)$ can be analytic at a point only if it satisfies the Cauchy-Riemann equations $u_x = v_y$ and $u_y = -v_x$ in a neighborhood of that point.

Step 2: Key Formulas and Approach
Express $f(z) = z - \bar{z}$ in real and imaginary parts using $z = x + iy$ and $\bar{z} = x - iy$: \[ f(z) = (x + iy) - (x - iy) = 2iy \] So $u(x, y) = 0$ and $v(x, y) = 2y$. Test C-R equations.

Step 3: Step-by-step Explanation

• Compute partial derivatives of $u(x, y) = 0$ and $v(x, y) = 2y$: \[ u_x = 0, \quad u_y = 0 \] \[ v_x = 0, \quad v_y = 2 \]
• Check C-R equation $u_x = v_y$: \[ 0 = 2 \quad \text{(False everywhere!)} \]
• Since $u_x \neq v_y$ at every point in $\mathbb{C}$, $f(z)$ does not satisfy the Cauchy-Riemann equations at any point. Hence, Reason R is true.
• Because satisfying C-R equations is a necessary condition for complex differentiability, $f(z)$ is nowhere differentiable, and thus nowhere analytic. Hence, Assertion A is false.

Step 4: Final Answer
Assertion A is false, but Reason R is true. Thus, Option (D) is correct.