Analytic function from u = x² - y²

Short answer

u=x²−y² is harmonic; f(z)=z²+ic′ with v=2xy+constant.

Determine the analytic function f(z) = u(x,y) + iv(x,y) if u(x,y) = x² - y².

u = x² − y².

u_x = 2x, u_y = −2y, u_xx = 2, u_yy = −2, so u_xx + u_yy = 0 ✓.

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

f′(z) = 2z − i·0 = 2z

Integrating: f(z) = z² + c.

z² = (x² − y²) + 2ixy. So v = 2xy + constant.

Check with C–R: v_y = 2x = u_x ✓ and v_x = 2y = −u_y ✓.

Answer: f(z) = z² + ic′ (that is, v = 2xy + c′).

✓ Answer checked by Claude Opus 5.5 (high effort)

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