Past exam of the mathematics course of the University of Cambridge / 2008 / ib / Paper 1 / 3C (Complex Analysis or Complex Methods)

For a regular smooth curve $z=z_j(t)$ passing through $z_0$, the complex tangent of its image is $d{w}_j/dt=f^{\prime }(z_0)z_j^{\prime }(t)$. When $f^{\prime }(z_0)\ne 0$, both tangents are multiplied by the same nonzero complex number. Multiplication rescales lengths by $|f^{\prime }(z_0)|$ and rotates every direction by $\arg f^{\prime }(z_0)$, so the difference of tangent arguments is unchanged. Hence the analytic function defines a conformal map at $z_0$: oriented angles, and therefore ordinary angles between the curves, are preserved. The nonzero derivative condition also ensures that the image tangents are nonzero.

Write $f=u+iv$ and $z=x+iy$. The Cauchy-Riemann equations give $u_x=v_y$ and $u_y=-v_x$. Thus the real Jacobian determinant isThis identity also holds at points where $f^{\prime }(z)=0$; there the Jacobian is zero and local angle preservation is not guaranteed.