The Cauchy-Riemann Equations

Complex Analysis · Axiom Academy

LESSON The Cauchy–Riemann Equations Two innocent-looking partial-derivative equations that say something deep: an analytic map can only rotate and scale. 1. Two Equations, One Right Angle Write with z = x + iy . The two equations couple the partials of u and v in a very particular pattern: read them as a statement about the two gradient vectors Stack them. The gradient of u is and the gradient of v is . That second vector is the first one rotated a quarter-turn : same length, turned . So Cauchy–Riemann is the single geometric demand rotated . 2. Why It Matters: a Rotation and a Scale Up close, any differentiable map of the plane acts on tiny arrows through its Jacobian. For f = u + iv that matrix is: Substitute Cauchy–Riemann ( u_x = v_y = a and v_x = -u_y = c ) and the four entries collapse into a scaled rotation : every column is the other turned . A matrix of that shape stretches every vector by the same factor and turns them all through the same angle — it preserves shape and right angles . In the picture a tiny square near a point is carried to its image. Under a Cauchy–Riemann map it stays a square — just rotated and resized . Toggle to a map that breaks the equations and the same square shears into a slanted parallelogram: angles ruined.

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