A couple of additions. Complex functions are often expressed using polar coordinates. Thus we might write
f(z) = u(r,t) + iv(r,t)
Where t is the polar angle theta.
The Cauchy Riemann equations now read:
(du/dr) = (-1/r) * (dv/dt)
(du/dt) = r * dv/dr
Also the Cauchy Riemann conditions are only necessary conditions for a function to be analytic. Continuity of the partial derivatives of u and v is the second condition required for sufficiency.