Prove: Let f(z) is analytic at z0 and (z0) 0. Then the under the mapping w = f(z) the tangent at z0 to any curve C passing through z0 is rotated through the angle arg (z0).
Proof. See Fig. 1. As a point moves from z0 to z0 + Δz along C, the image point traces out C' in the w plane, going from w0 to w0 +Δw. Assume that the curves C and C', are defined parametrically with t as the parameter. Then corresponding to the path z = z(t) [or x = x(t), y= y(t)] in the z plane, we have the path w = w(t) [or u = u(t), v = v(t)] in the w plane.
The derivatives dz/dt and dw/dt represent tangent vectors to corresponding points on C and C'.
At points z0 and w0, we have
Equation 1) then becomes
which is what we wished to prove.