Key difference between two real and single complex variable?

[SVN]

Notion of differentiability (analyticity) for function of complex variable is normally introduced and illustrated by comparison with function of single real variable.

It is stated that there are infinite number of ways to approach any given point of complex plane where function is defined, not just «left» anf «right» for the case of real variable. That is where necessity for having the same limit no matter which direction is regarded comes into play and the next step is to deduce the Cauchy-Riemann equations from this necessity.

Would not it be more appropriate to compare the case of complex variable with case of two real variables? Here one has x — y plane and different directions as well, but no eqivalent of C.-R. equations, one starts considering partial derivatives instead. Why is that? What is the qualitative difference between pairs (a; b) and (a; ib)? And what about split-complex numbers, where i^2 = +1 from the viewpoint of differentiability of their functions?

Sorry, if this thread is redudant and my question has already been answered in another one. Just point me to the latter in this case, I failed to find it myself.

 

[stevendaryl]

Would not it be more appropriate to compare the case of complex variable with case of two real variables? Here one has x — y plane and different directions as well, but no eqivalent of C.-R. equations, one starts considering partial derivatives instead. Why is that? What is the qualitative difference between pairs (a; b) and (a; ib)? And what about split-complex numbers, where i^2 = +1 from the viewpoint of differentiability of their functions?

An analytic function of a single complex number is a special case of a function of two real numbers. It's a special case that is much easier to deal with, mathematically.

If you have a function $F(x,y)$ of two real variables, then $\dfrac{\partial F}{\partial x}$ and $\dfrac{\partial F}{\partial y}$ are unrelated. But in the case of a single complex variable: $z = x + i y$, we have the constraint $\dfrac{\partial F}{\partial x} = -i \dfrac{\partial F}{\partial y}$. So we don't need partial derivatives, we can just write $\dfrac{dF}{dz}$. There are lots of mathematical advantages to using complex numbers, and lots of theorems that work for complex functions that don't work (with additional assumptions) for general functions of two variables. For example, complex numbers have an associated product that has an inverse: $z \cdot z^{-1} = 1$.

The split complex numbers and the regular complex numbers can both be thought of as special cases of a more general kind of algebra, called "Clifford algebras". They have some of the nice features of complex numbers, but not all of them (in particular, the split complex numbers don't have a unique inverse, I don't think).

 

[SVN]

Thanks a lot for reply, it has become much clearer now.

I'd like to be sure I got your idea. Can we consider space of complex numbers to be effectively space ℝ² equipped with special constraint coupling ℝ and ℝ? That means complex number to be a single entity (in some cases one can find it convenient to deal with it as if it would be two unrelated entities, i. e. normal ℝ², but the coupling is not to be forgotten at the end). Please, correct me if I am wrong.

But the thing which still puzzles me is where this coupling come from? As far as I know, complex space is essentially a vector space over field of real numbers. But the very idea of vector space is based on fact the basis vectors, no matter which proper basis we use, are linearly independent of each other (they can always be ortogonalised). How does this independence agree with the mentioned coupling (sorry if my questions are naïve, my background in mathematics is not exactly strong).

 

[FactChecker]

Multiplication of complex numbers provide a rotation operation in C that regular R² vector spaces do not have. That is a very important geometric operation to have. There is so much in physics and math that you want to represent by a local scaling and twisting. With that definition of complex multiplication, the complex derivative can mimic the property in R where the derivative is a local multiplication by a single number (dx * slope in R and (scale+rotate) dz in C). The existence of a complex derivative gives analytic functions surprising properties and lead to powerful results.

 

[SVN]

Thank you for explanation. That answers my question.