Analytic function from u = 3x²y - y³

u = 3x²y − y³.

Check that u is harmonic

u_x  = 6xy          u_xx = 6y
u_y  = 3x² − 3y²    u_yy = −6y

u_xx + u_yy = 0, so u is harmonic and an analytic f with Re f = u exists.

Milne-Thomson method

f′(z) = u_x − i u_y. Put x = z, y = 0:

f′(z) = 0 − i(3z² − 0) = −3i z²

Integrate: f(z) = −i z³ + c.

Find v

z³ = (x³ − 3xy²) + i(3x²y − y³), so

−i z³ = (3x²y − y³) − i(x³ − 3xy²)

Real part = 3x²y − y³ = u ✓, and v = −(x³ − 3xy²) = 3xy² − x³ + constant.

Check with C–R: v_y = 6xy = u_x ✓ and v_x = 3y² − 3x² = −u_y ✓.

Answer: f(z) = −i z³ + c, with v = 3xy² − x³ + constant.