Here you go.
Proof To Cauchy Riemann Equations Polar Coordinates
Suppose f = u + iv is analytic in a domain D. Show that the cauchy-Reimann equation in polar coordinates are:
r(2u/2r) = (2v/2Theta)
and
r(2v/2r) = -(2u/2Theta)
****Each '2' is representing 'partial of' -- NOT ACTUALLY THE #2 ******
Here you go.
Proof To Cauchy Riemann Equations Polar Coordinates