Discussion
Consider two real functions
\[ \begin{aligned} U(x, y) &= xy(x^{2} - y^{2}) , \\ V(x, y) &= ax^{4} + by^{4} + cx^{2}y^{2} + k , \end{aligned} \]
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 \(a \times b \times c\)?
Answer: D
The product is \(a \times b \times c = \frac{3}{32}\).
\(U = x^{3}y - xy^{3}\). For \(U + iV\) to be analytic, the Cauchy-Riemann equations \(U_{x} = V_{y}\) and \(U_{y} = -V_{x}\) must hold:
\[ \begin{aligned} 3x^{2}y - y^{3} &= 4by^{3} + 2cx^{2}y \\ x^{3} - 3xy^{2} &= -4ax^{3} - 2cxy^{2} \end{aligned} \]
Matching coefficients: \(2c = 3\), \(4b = -1\) and \(4a = -1\), so \(a = b = -\frac{1}{4}\) and \(c = \frac{3}{2}\). Then
\[ abc = \frac{1}{16} \times \frac{3}{2} = \frac{3}{32} \]
The constant \(k\) drops out of every derivative. The other values do not satisfy both equations; 1/8, for example, would need \(c = 2\).
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