So the way I understand complex differentiability and its requirement that the partial derivatives satisfy the Cauchy-Riemann Equations is that we would really like ℂ to have the same nice property as ℝ, that is to say we would really like the derivative to be a linear operator which is itself an element of the field. Thus we put this restriction on our complex derivative and voilà, its Jacobian matrix turns into an element of that special subfield of the ring M(ℝ)2x2, isomorphic to ℂ.(adsbygoogle = window.adsbygoogle || []).push({});

Now this restriction allows us to derive all sorts of nice properties about functions which are complex differentiability, the big one being analyticity. On the other hand differentiable functions from ℝ^2 to ℝ^2 seem to get on reasonably well without restrictions on the components of their Jacobian matrices, and I understand that adding the Cauchy-Riemann restriction wouldn't get us anywhere interesting in this case. The question is what would happen if we dropped the Cauchy-Riemann Equations for ℂ? Would we just end up with ℝ^2? Clearly the same functions would be differentiable, but with ℂ we still have a field and not just a vector-space, and thus I feel like a theory of this rougher differentiability would still be distinct from that for ℝ^2.

This makes me feel that something deeper or more subtle is going on, something to do with linear algebra I imagine, something maybe to do with the fact that all fields are one dimensional vector-spaces over themselves, I'm not really sure, I can't quite find the correct perspective. Please feel free to ramble, elaborate, digress and/or school me about complex analysis in general or about any misconceptions I may have.

Edit: Now that I think a bit more about it I imagine we would have some problems taking the limit of the difference quotient, since dropping the Cauchy-Riemann Equations would cause the limit to be different depending on from which direction we approach the point, which is allowed in the ℝ^2 case. And if I'm remembering correctly, in ℂ it's enough to show that approaching the limit horizontally and vertically( with h times i instead of h ) and showing they're equal is enough to know that approaching it from any direction will also be equal, and thus equivalent to the C-R Equations. I can't remember though if this is true in the ℝ^2 case, I think not, but somehow this doesn't matter as long as the partial derivatives are continuous, so is it the multiplicative structure of ℂ that causes this?

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# Better Understanding of Complex Differentiability

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