Question

Consider two real functions \[ U(x,y) = xy(x^2 - y^2), \] \[ V(x,y) = ax^4 + by^4 + cx^2y^2 + k, \] where $k$ is a real constant and $a, b, c$ are real coefficients. If $U(x, y) + i V(x, y)$ is analytic, then what is the value of $a \times b \times c$?

• A. $\dfrac{1}{8}$
• B. $\dfrac{3}{28}$
• C. $\dfrac{5}{36}$
• D. $\dfrac{3}{32}$

Hint:

For analytic functions, always use the Cauchy-Riemann equations to find relationships between the coefficients in $U(x, y)$ and $V(x, y)$.

Answer 1

The Correct Option is D

For the function $U(x,y) + i V(x, y)$ to be analytic, it must satisfy the Cauchy-Riemann equations: \[ \frac{\partial U}{\partial x} = \frac{\partial V}{\partial y}, \frac{\partial U}{\partial y} = - \frac{\partial V}{\partial x}. \] Using these equations, we can derive the values for the coefficients $a$, $b$, and $c$ by matching terms from the partial derivatives of $U(x, y)$ and $V(x, y)$. After solving, we find that the value of $a \times b \times c$ is $\dfrac{3}{32}$. Thus, the correct answer is (D) $\dfrac{3}{32}$.