Deriving Cauchy-Riemann

Complex Analysis · Axiom Academy

EXAMPLE Deriving Cauchy-Riemann Force the complex derivative to agree from every direction — and the equations fall out Write where z = x + iy . Suppose f is differentiable at z_0 . The single limit must give the same value no matter how the complex step h approaches 0 . By comparing two approaches — along the real axis and along the imaginary axis — find the conditions u and v must satisfy. You derived the Cauchy-Riemann equations from a single idea: a complex derivative exists only if the limit is direction-independent. Imaginary-axis approach: dividing by and using gives . Equating the two: u_x = v_y and u_y = -v_x — the Cauchy-Riemann equations. Bonus formula: the derivative can be computed either way, . Two directions were enough here; with continuous partial derivatives, the Cauchy-Riemann equations guarantee the limit agrees from every direction — that local linear map is genuine complex multiplication (a rotation and scaling), not an arbitrary 2×2 matrix.

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