x - Aerospace Computing Lab - Stanford University

Cauchy-Riemann Equations
• Extension of rational numbers to solve x 2 = 2:
Define w = a + √ 2b where a, b are rational.
Derivative of f(w):
∂f
∂w
= ∂f
∂a = √ 2 ∂f
∂b .
• Extension of real numbers to solve x 2 = −1:
Define z = x + iy, where x, y are real.
Derivative of f(z):
∂f
∂z
= ∂f
∂y
. Define f(z) = u(z) + iv(z),
∂u
∂x
= ∂v
∂y ,
= i∂f
∂x
∂u
∂y
= −∂v
∂x .
Complex number really just one number.
– Joaquim Martins – January 12, 2000 – 4