Verifying C-R for z²

Complex Analysis · Axiom Academy

Working the Cauchy-Riemann test on f(z) = z^2 , then reading off the derivative Show that f(z) = z^2 satisfies the Cauchy-Riemann equations everywhere, then use the C-R derivative formula to compute f'(z) . Nice work — you ran the full Cauchy-Riemann check on f(z) = z^2 and recovered its derivative without ever taking a complex limit. Split first: with z = x + iy , every f(z) becomes . Here u = x^2 - y^2 and v = 2xy . The two C-R equations: analyticity requires u_x = v_y and u_y = -v_x . Watch that minus sign — it is what separates z^2 from a non-analytic function like . Continuity seals it: the partials are polynomials, so they are continuous on all of . C-R holding plus continuous partials means f is differentiable everywhere. The derivative falls out: . The alternative formula gives the same 2z — a built-in check. Result: , exactly the power rule, now justified through C-R. This same six-line routine — split, differentiate, check both equations, confirm continuity, read off — verifies analyticity and computes the derivative for any candidate function.

This is the written version of the interactive lesson above. See the full Complex Analysis course.