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Complex Analysis

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/0:
In the complex derivative,



Can 'h' be anything?

This looks a bit like the directional derivative formula in :



But yet there is a 'unique' ? I would have thought should vary depending on the direction you approach.

(This question of course leads to how the Cauchy-Riemann equations came about)

Thanks

zzdfa:
Remember the definitions: a function is differentiable only if that limit exists.
If you get 2 different values depending on which direction you approach, then the limit does not exist and hence the function is not differentiable. if it -is- unique, then we say it is differentiable.



for example f(z)=Re(z)+2Im(z)= is not complex differentiable at 0 because the limit doesn't exist.

/0:
But if you think of the function as a 'contour map' on the x-y plane ()... if every point in a domain has derivatives from every side equal, then surely the function must be constant?

zzdfa:
It is true that if the directional derivatives of f:R^2->R at each point are the same for every direction, then f is constant.  (because if directional derivative is the same in all directions it implies the gradient is 0 at that point. if the gradient at 0 at every point then f is constant)

However you can't think of a complex function C->C as a contour map, remember the range of the function the complex numbers, so the graph is a surface in C^2, or R^4.


just try a simple example, check that if f(z)=z^2 then  

/0:
Ah, ok thanks zzdfa =)

Another Q, if you approach a point along the imaginary axis then:



Is the from the in the limit?

Since

It seems that if you set , then it essentially becomes the first limit above:
Since ( is the same as right?)

So is it really ok to take the out?

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