Derive Cauchy-Riemann equations (Cartesian)

Analytic function

A function f(z) is analytic at a point z₀ if it is differentiable at z₀ and at every point in some neighbourhood of z₀. It is analytic in a region if it is differentiable at every point of that region.

Cauchy–Riemann equations

Theorem. If f(z) = u(x, y) + i v(x, y) is analytic in a region, then at every point of it

∂u/∂x = ∂v/∂y and ∂u/∂y = −∂v/∂x.

Derivation

Since f is differentiable at z = x + iy, the limit

f′(z) = lim_{δz→0} [f(z + δz) − f(z)]/δz

exists and is the same along every path by which δz → 0. Here δz = δx + i δy.

Path 1: along the real axis (δy = 0, so δz = δx → 0):

f′(z) = lim [u(x + δx, y) − u(x, y)]/δx + i [v(x + δx, y) − v(x, y)]/δx = ∂u/∂x + i ∂v/∂x. ... (1)

Path 2: along the imaginary axis (δx = 0, so δz = i δy → 0):

f′(z) = lim [u(x, y + δy) − u(x, y)]/(i δy) + i [v(x, y + δy) − v(x, y)]/(i δy) = (1/i) ∂u/∂y + ∂v/∂y = ∂v/∂y − i ∂u/∂y. ... (2)

The two values must be equal. Equate real and imaginary parts of (1) and (2):

∂u/∂x = ∂v/∂y
∂v/∂x = −∂u/∂y

These are the Cauchy–Riemann (C–R) equations. ∎

Also f′(z) = u_x + i v_x = v_y − i u_y.

Sufficient condition. If u, v, and their first partial derivatives are continuous in a region and satisfy the C–R equations there, then f(z) = u + iv is analytic in that region.