Vrftsjtryk

Are the derivative rules the same? I mean I have seen the definition, etc., it is the same as in the real case, but I cant stop thinking of the complex plane as $R^2$ and functions as vector fields. So all I can think of are partial derivatives and total derivatives as in multivariable calculus.

My intuition goes the other way.

So can I use the same rules without fear when differentiating? For example, with $ f(z) = frac{1} {z-i} $ - can I use the quotient rule. And is there any intuitive explanation why I can do that? Is it because I have just one variable? So it doesn't matter if that variable has a corresponding vector?

When I saw the definition of the limit of the derivative, the first thing I thought was "You cannot divide by $h$ since $h$ is a vector."

For the most part, the rules of differentiation with complex functions are the same as with real functions, just treating the number $i$ as a constant and letting $i^2 = -1$ where it comes up.

The sum rule, product rule, quotient rule, chain rule, etc., can be relatively easily derived as in standard calculus.

The rules for differentiating specific functions, i.e. how $(sin(z))' = cos(z)$, can also be easily derived to some extent. Arguably even easier than in the real case since you have relations like $cos(z) = (e^{iz} + e^{-iz})/2$ that can also be derived.

You could even make the argument that the reason why that the rules are the same is not because of some special property that holds in going from $mathbb{R}$ to $mathbb{C}$, but rather because $mathbb{R}$ is a subset of $mathbb{C}$, i.e. the same rules that apply for the complex system generally hold for the reals, except that some special properties are introduced by this "narrowed focus" on the reals, as it were. In other words, it'd be the reverse - that the properties of the derivatives hold in the reals as they do in the complex numbers, because they hold in the complex numbers and that includes the reals.

You can also think, if you prefer, as complex numbers as vectors, and complex-valued functions as functions of real variables. That is to say, you can think of a complex-valued function $f(z)$ as

$$f(z) = u(x,y) + iv(x,y)$$

where $u,v$ are real-valued functions of two variables, $x,y$, analogous to how complex numbers are of the form $z = x + iy, x,y in mathbb{R}$, which brings with it a lot of the same intuitions as you would find with vectors and $mathbb{R}^2$ and all as you were mentioning.

However, the math involved can be fruitlessly messy if you do things like that, so it will be illustrative and easier on you to fixate on developing a different intuition for complex analysis: you can't use the same intuitions for every field of math after all. Every field, every number system has its own quirks and features, and so too you should develop individual intuitions for each, instead of trying to tie everything back to older stuff you learned, which can be, at best, cumbersome. At worst, you can be led down the wrong path.

In my experience, it is helpful to interweave between interpretations of complex numbers as vectors or as numbers as you feel necessary. Personally, I got through my own complex analysis course by thinking of them almost entirely as numbers in the complex plane: vectors don't totally mesh well with me for some reason. My professor, however, loved to use analogies with vectors - which, I admit, were oftentimes illustrative. So it's best to find an interpretation that fits well at the time and in the given context, but to never settle firmly on one. The grass is always greener and whatnot.

In light of that number interpretation, I suppose a lot of the same results may follow because, indeed, a complex number is indeed just one number - $z$ is just one variable. It can be decomposed as a function of $x,y$ as above, but overall it is still just one singular number. Of course, it can also be treated as vector too, where appropriate or necessary.