Geometrical Interpretation of Cauchy Riemann equations? Differentiation has an obvious geometric interpretation, and the Cauchy Riemann equations are closely linked with differentiation.
Do the Cauchy Riemann equations have a geometric interpretation?

Example of action of a derivative of complex functions. If f(z)=z^2, f'(z)=2z.
For example, at z=1+i, the derivative is f'(1+i)=2(1+i). So if you apply this to something like 0.01, a differential along the horizontal axis in the complex plane, you get 
2(i+1)(0.01)=0.02(1+i)=0.02+0.02i. The "2" in 2(i+1) stretches 0.01 and the 1+i turns it 45 degrees, since 0.02/0.02=1 and arctan(1)=45 degrees.  So you can approximate (1.01+i)^2 as 
(1+i)^2+2(i+1)(0.01) as a first step. This has the form f(z)+f'(z)dz roughly speaking. Or in words, value of function+amplitwist * differential. (Amplitwist is from Visual Complex Analysis by Needham)
