Project ForgeEngineering Mathematics
Calculus
GATE 2019
Hard
A harmonic function is analytic if it satisfies the Laplace equation. If u(x,y) = 2x² − 2y² + 4xy is a harmonic function, then its conjugate harmonic function v(x,y) is
- 4xy − 2x² + 2y² + constant
- B4y² − 4xy + constant
- C2x² − 2y² + xy + constant
- D−4xy + 2y² − 2x² + constant
Explanation
Cauchy-Riemann: ∂v/∂y = ∂u/∂x = 4x+4y, so v = 4xy + 2y² + f(x). Also ∂v/∂x = −∂u/∂y = 4x−4y. Differentiating v: ∂v/∂x = 4y + f′(x). Equating: 4y+f′(x) = 4x−4y → f′(x) = 4x−8y, but f depends only on x, so matching the x-only part gives f′(x)=−4x → f(x)=−2x². So v = 4xy + 2y² − 2x² + constant.
Source: GATE ME 2019 Session 1 (Set 1) (2019)
This question is part of a 15-day revision cycle — sign in and Project Forge will bring it back the day before you'd forget it.