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:
uvuv
-- = -- ,   -- = - --
∂xyyx