Why z̄ Isn't Analytic

Complex Analysis · Axiom Academy

The simplest map that fails to be complex differentiable — anywhere. The conjugation map just reflects each point across the real axis — about as simple as a function gets. Yet it is not complex differentiable at a single point . Show this two ways: directly, from the limit definition of the derivative; using the Cauchy–Riemann equations. The geometry behind the failure As a real map of the plane, sends — a reflection across the real axis. Its matrix is , with determinant -1 : it reverses orientation . Complex differentiability demands that, up close, f act like multiplication by one complex number — a rotation-and-scale, which keeps the little orange angle turning the same way (it is conformal , det ). Reflection flips that angle from counter-clockwise to clockwise (red): it is anti-conformal. No single complex number multiplies like a flip, so no derivative can exist. Nice work — you proved is nowhere complex differentiable two independent ways and saw why. Directions disagree: the difference quotient collapses to , which tends to +1 along the real axis but -1 along the imaginary axis. Cauchy–Riemann fails: with you get at every point. Geometry: conjugation is a reflection (orientation-reversing), not a rotation — so it can't be "multiplication by a complex number." Red flag: any formula that uses explicitly is suspect for differentiability. That's why is differentiable only at z = 0 .

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