Stream function and complex potential

The complex potential is f(z) = φ + iψ, where f is analytic and φ, ψ satisfy the Cauchy–Riemann equations φ_x = ψ_y, φ_y = −ψ_x.

Check that φ is harmonic

φ = 3x²y − y³
φ_xx = 6y,  φ_yy = −6y  → φ_xx + φ_yy = 0

Stream function

ψ_y = φ_x = 6xy   → ψ = 3xy² + g(x)
ψ_x = 3y² + g'(x) = −φ_y = −(3x² − 3y²) = 3y² − 3x²
g'(x) = −3x²  → g = −x³
ψ = 3xy² − x³ + c

Complex potential

Since z³ = (x³ − 3xy²) + i(3x²y − y³), we have φ = Im(z³) and ψ = −Re(z³):

f(z) = φ + iψ = (3x²y − y³) + i(3xy² − x³) = −i z³ + c

Check: −i z³ = −i(x³ − 3xy²) + (3x²y − y³) ✓.

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