Question

If the function \( f(z) = x^2 + axy + by^2 + i(cx^2 + dxy + y^2) \) is analytic, then the values of \( a, b, c, \) and \( d \) are given by:

• A. \( a = 2, b = -1, c = 1, d = -2 \)
• B. \( a = 2, b = 1, c = -1, d = 2 \)
• C. \( a = 2, b = 1, c = 1, d = 2 \)
• D. \( a = 2, b = -1, c = -1, d = 2 \)

Hint:

Check analyticity by applying the Cauchy-Riemann equations and verifying their consistency.

Answer 1

The Correct Option is D

Analyticity of \( f(z) \) necessitates satisfying the Cauchy-Riemann equations:

\( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \)

Applying these equations to the function reveals the constants: \( a = 2, b = -1, c = -1, \) and \( d = 2 \).