What are the Cauchy-Riemann equations and their geometric interpretation?

[arshavin]

Is there a geometric meaning for the derivative of a complex valued function, or any other motivation for the derivative?

 

[Dick]

Do you mean the Cauchy-Riemann equations? They just say that if you calculate the directional derivative along any line through a point you'll get the same answer regardless of direction. Write down the directional derivative of a function along the real direction and the imaginary direction and set them equal. You'll get CR.