Geometrical Interpretation of Cauchy Riemann equations?

Question

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?

Answer 1

Yes, but in four dimensions, since the domain and the codomain is a two dimensional vector space over reals. Hint: partial derivatives are directional slopes.

There is a book called visual complex analysis that could be useful.

Answer 2

The Cauchy-Riemann equations are conditions that are necessary and sufficient for a complex function to possess a derivative at a given point in the complex plane. For a real function, the derivative is defined as a limit as h approaches zero, and the limit must be the same regardless of whether h approaches zero from the left or from the right, in order for the derivative to exist. In order for a complex function, f(z), to be differentiable, the derivative, f'(z), must be the same regardless of the direction from which the point is approached. In other words, h can approach zero along the real axis or along the imaginary axis, and both answers must match in order for the derivative to exist.