Harmonic function and conjugate

Short answer

u_xx = 6x, u_yy = -6x, so u_xx+u_yy=0. Hence u is harmonic (Laplace equation satisfied). It is Re(2z+z^3).

Show that the function u(x, y) = 2x + x³ − 3xy² is harmonic.

A function is harmonic if it satisfies Laplace's equation u_xx + u_yy = 0.

u = 2x + x³ − 3xy²
u_x = 2 + 3x² − 3y²     u_xx = 6x
u_y = −6xy              u_yy = −6x
u_xx + u_yy = 6x − 6x = 0

All second partial derivatives are continuous, so u is harmonic. (u = Re(2z + z³).)

✓ Answer checked by Claude Opus 5.5 (high effort)

Related questions