A harmonic function is analytic if it satisfies the Laplace

A harmonic function is analytic if it satisfies the Laplace equation. If u(x, y) = 2x² – 2y² + 4xy is a harmonic function, t hen its conjugate harmonic function v (x, y) is

1. 4y² – 4xy + constant
2. 4xy – 2x² + 2y² + constant
3. 2x² – 2y² + xy + constant
4. – 4xy + 2y² – 2x² + constant

u(x, y) = 2x² – 2y² + 4xy
As harmonic function is analytic therefore,
u_x = V_y
u_y = – V_x
u_x = 4x + 4y

	δv	V_y = 4x + 4y
	δy

V = 4xy + 2y² + f(x)
Now

	δv	V_y + f'(x) = 4y - 4x
	δy

f'(x) = – 4x
f(x) = – 2x²2 + C
So, conjugate harmonic function v(x, y) is,
V = 2y² – 2x² + 4xy + C