Pair of
differential equations satisfied by any
holomorphic (
complex analytic)
function f:
C ->
C.
We can write f(z) = u(z) + iv(z) for real functions u,v, and then write z=x+iy. When
f(z+h)-f(z)
lim -----------
h->0 h
exists, the
Cauchy-
Riemann equations state that the
partial derivatives satisfy:
∂u ∂v ∂u ∂v
-- = -- , -- = - --
∂x ∂y ∂y ∂x