Harmonic function and conjugate

Short answer

u satisfies Laplace's equation. Harmonic conjugate v = x³ − 3xy² + c, analytic function f(z) = i z³ + c'.

Prove that u = y³ − 3x²y is a harmonic function. Find its harmonic conjugate and hence construct the corresponding analytic function w = f(z).

u = y³ − 3x²y
u_x = −6xy     u_xx = −6y
u_y = 3y² − 3x²   u_yy = 6y
u_xx + u_yy = 0

So u satisfies Laplace's equation and is harmonic.

Use the Cauchy–Riemann equations v_y = u_x and v_x = −u_y.

v_y = −6xy   → v = −3xy² + g(x)
v_x = −3y² + g'(x) = −u_y = 3x² − 3y²
g'(x) = 3x²  → g = x³
v = x³ − 3xy² + c

f(z) = u + iv = (y³ − 3x²y) + i (x³ − 3xy²) + ic

Since i z³ = i(x³ + 3ix²y − 3xy² − iy³) = (y³ − 3x²y) + i(x³ − 3xy²), we get

Answer: v = x³ − 3xy² + c and w = f(z) = i z³ + c'.

✓ Answer checked by Claude Opus 5.5 (high effort)

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