Show u = x²+2xy-y² harmonic; find v

Show that u is harmonic

u = x² + 2xy − y².

u_x = 2x + 2y     u_xx = 2
u_y = 2x − 2y     u_yy = −2

u_xx + u_yy = 0, and the second derivatives are continuous. So u is harmonic.

Find v (C–R equations: v_y = u_x, v_x = −u_y)

Integrate v_y = 2x + 2y with respect to y:

v = 2xy + y² + g(x)

Then v_x = 2y + g′(x) must equal −u_y = −2x + 2y. So g′(x) = −2x and g(x) = −x² + c.

Answer: v = 2xy + y² − x² + c

The analytic function

f′(z) = u_x(z, 0) − i u_y(z, 0) = 2z − 2iz = 2(1 − i)z

f(z) = (1 − i) z² + C.

Check: (1 − i)(x² − y² + 2ixy) = (x² − y² + 2xy) + i(2xy − x² + y²). The real part is u ✓ and the imaginary part is v ✓.