Question

For which values of \(k\) does the function \(f(z)=\left(x^3-3xy^2\right)+i\left(kx^2y-y^3\right)\) become analytic?

• A. 0
• B. 1
• C. 2
• D. 3

Hint:

A complex function \(f(z)=u+iv\) is analytic when it satisfies the Cauchy-Riemann equations: \(u_x = v_y\) and \(u_y = -v_x\). We match the partial derivatives of the real and imaginary parts to find k.

Answer 1

The Correct Option is D

Concept:
A complex function \(f(z)=u+iv\) is analytic when it satisfies the Cauchy-Riemann equations: \(u_x = v_y\) and \(u_y = -v_x\). We match the partial derivatives of the real and imaginary parts to find k.

Step 1:
Identify the parts. The real part is \(u = x^3 - 3xy^2\) and the imaginary part is \(v = kx^2y - y^3\).

Step 2:
Compute the partial derivatives. For u: \(u_x = 3x^2 - 3y^2\) and \(u_y = -6xy\). For v: \(v_x = 2kxy\) and \(v_y = kx^2 - 3y^2\).

Step 3:
Apply the first Cauchy-Riemann equation \(u_x = v_y\): \[3x^2 - 3y^2 = kx^2 - 3y^2.\] Comparing the \(x^2\) terms gives \(k = 3\).

Step 4:
Check the second equation \(u_y = -v_x\): \(-6xy = -2kxy\), so \(6 = 2k\), again \(k=3\). Both conditions agree.

Answer: Option (4) — 3.