What's the geometrical interpretation of the derivative of a ...

[Melanie]

The geometrical interpretation of the derivative of a real function is simply the slope of the line tangent to the function at that point.  What about complex functions?
Uh, functions of a complex variables, okay?  F(z), where z is the complex number x + iy.

[federox]

Complex functions being composite functions?

If so, where there are multiple functions of defined intervals, then it's the same.  The derivative is the slope of any given 'piece' at a point.  To know which 'piece' to use, you look at the intervals given in the composite function.  Wherever your point fits in, that's the function you use to find derivative of.

[Sandstorm]

I'm going to take a stab at this just because no one else has.

My belief is this: since complex function derivatives are taken the same exact way that real function derivatives are taken (keeping in mind that the definition of the derivative is derived from the slope of the secant line), the geometric interpretation is very similar, with the only difference being that they can take on complex values, and therefore must be plotted in the complex plane.

[TI]

A geometric interpretation of the derivative of a complex function at a particular point is, well, probably not best said in terms of tangency. This is because the graph of a complex function lies in 4-space. (I checked my two complex analysis books, and googled a little too and found nothing on a geometric interpretation like this.)
A more relevant interpretation would be that of linear approximation. The function
L(z) = f'(c)(z - c) + f(c)
gives a "suitable" linear approximation to f(z) near z = c.
I'm not sure if that is some sort of tangent plane in 4-space or what.

Hope this helps.

[ricki]

I am taking the following from Markushevich's Theory of Functions, Volume 1, Chapter 8 (it is out of print and I got a copy cheap years ago. Very hard to find now even in libraries.) The chapter is 8 pages long, so I will just briefly describe.

Geometric Interpretation of Arg f'(z) [remember that the argument is the angle part in the polar representation of a complex number]: let w=f(z) be a complex function. Let Z be the plane of the domain and W be the plane of codomain. Suppose f(z) has a non-zero derivative at some point z0. Let l be a curve in the Z-plane that passes through z0. Let L be its image in W-plane. Let S be the tangent ray of l at the point z0, and T be the tangent ray of L at the point f(z0), then the difference in inclination of T and S is Arg f'(z0) (the inclination is defined as the angle formed between the ray and the positive real ray).

In other words, if f(z) is a differentiable function at z0 with non-vanishing derivative, then locally it is invertible. So given a small disk D about z0, f(z) maps it 1-to-1 to some small disk D'. Arg f'(z0) represents how much the disk D' is rotated relative to the disk D.

Geometric interpretation of |f'(z)|: Using Markushevich's words, |f'(z)| is a linear magnification ratio. Thinking the tangent rays S and T from above as representing velocity vectors, |f'(z0)| gives how much faster T is relative to S.

Using the disk consideration from above, the quantity |f'(z)| represents how much the disk D' is magnified relative to the disk D.

These geometric interpretations are the forefathers of the branch of mathematics now called conformal geometry.