Complex Derivative and Real Multivariable Derivative

[cathode-ray]

Hi!

I'm studying complex analysis and I don't quite understand why the complex derivative was defined the way it is, and how it is related to the definition of the derivative of a real multivariable function.

 

[lurflurf]

Which equivalent defiition are you referring to? The usual definition is analogous to the real case.
f'(z)=lim [f(z+h)-f(z)]/h

One possible source of confusion (though this is also standard) is we require the limit to approach the same value in all possible ways, which can be a stong condition. Sometimes it is helpful to define and use derivatives which are defined by some restricted direction of approach. We must never confused such restricted derivatives with the derivative, but often we may conclude from the existence and behavior of such restricted derivatives that the derivative exists. This is what is done when we conclude from the Cauchy–Riemann equations that the derivative of some function exists.

 

[cathode-ray]

That is a point of confusion to me too. Why is it helpful to have that restriction in the complex case?
The other thing that makes me some confusion is that we can see a complex function as a function from R^2->R^2. If we think this way doesn't the complex derivative should be given through the Jacobian matrix, because of the multivariable derivative definition?

I don't know if this question has logic, but I start thinking about this and I can't figure it out. Maybe I didn't really understand the multivariable derivative definition yet :(

 

[lurflurf]

In any limit we allow certain directions, and for the limit to exist we require that it approach the same value for all allowed directions. What varies is what directions we allow. The complex differentiation case can appear strange because we allow so many more directions than the single variable case andreject funnctionwich we would accept in the multivarible case. What charaterizes the complex variable theory is not just that we don't get freaked out by sqrt(-1), complex variables is concerned with extraordinarily well behaved functions. When considering a complex function as
f:R^2->R^2
we must recall that f is limited in how it acts by its compatability with complex numbers. f:(x,y)->(x^2-y^2,2xy) is good since it is z^2 while f:(x,y)->(x^2+y^2,2) is very bad because it is not treating z as a single thing. One common approach is to say a function of two variables is like a function of z and z* (the complex conjugate of z) and for the derivative to exist the function can depend upon z* only in ways that depend upon z. In this formulation the Cauchy–Riemann equations take the form f_z*=0. The Jacobian will give the derivative when it exists, but the Jacobian can exist when the derivative does not and gives other stuff.

 

[lavinia]

The other thing that makes me some confusion is that we can see a complex function as a function from R^2->R^2. If we think this way doesn't the complex derivative should be given through the Jacobian matrix, because of the multivariable derivative definition?

yes. But the Jacobian of an analytic function can not be arbitrary. It must be the matrix form of a complex number. So the matrix must be a rotation followed by a scale expansion.

Multiplication by i for instance is a rotation counter clockwise by pi/2. Its matrix form is

0 -1
1 0

 

[cathode-ray]

I finally got the idea behind the complex derivative.

With respect to the Jacobian matrix I didn't knew that detail you posted lavinia. I googled it and I found a pdf that helped me to clarify my mind(http://www.brynmawr.edu/physics/DJCross/docs/papers/jacobian.pdf ).

Really thanks for your help!

 

[mathwonk]

the derivative is a linear approximation. the real derivative is a real linear linear approximation, and the complex derivative is required to be a complex linear approximation. It follows that the function has a complex derivative if and only if it has a real derivative, and that real linear approximation is also complex linear. This extra condition is called the cauchy riemann equations.

 

[Landau]

Ususal differentiability of the function f:R^2\to R^2 is weaker than complex differentiability of f:C\to C. Basically, the latter is precisely the Cauchy-Riemann equations (as mathwonk said), and this is what makes complex analysis so different from real (R^2) analysis.

You might want to look at this answer, where I tried to explain this more precisely.