GATE 2023 PH – Question 43
$U=xy(x^2-y^2)$ and $V=ax^4+by^4+cx^2y^2+k$. If $U+iV$ is analytic, find $abc$.
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Correct answer: (D) $3/32$
Explanation
For $U+iV$ to be analytic, the real and imaginary parts must satisfy the **Cauchy-Riemann equations**:
$$\frac{\partial U}{\partial x}=\frac{\partial V}{\partial y},\qquad\frac{\partial U}{\partial y}=-\frac{\partial V}{\partial x}.$$
With $U=xy(x^2-y^2)=x^3y-xy^3$:
$$U_x=3x^2y-y^3,\qquad U_y=x^3-3xy^2 .$$
**Integrate $V_y=U_x$** with respect to $y$:
$$V=\tfrac32x^2y^2-\tfrac14y^4+g(x).$$
**Use $V_x=-U_y$:**
$$V_x=3xy^2+g^\prime(x)=-x^3+3xy^2\;\Rightarrow\;g^\prime(x)=-x^3\;\Rightarrow\;g(x)=-\tfrac14x^4+k .$$
$$V=-\tfrac14x^4-\tfrac14y^4+\tfrac32x^2y^2+k .$$
Comparing with $V=ax^4+by^4+cx^2y^2+k$: $a=b=-\tfrac14$ and $c=\tfrac32$:
$$abc=\left(-\tfrac14\right)\left(-\tfrac14\right)\left(\tfrac32\right)=\frac{3}{32}\quad(\text{option D}).$$