# How do the differentials you get from complex functions behave?

I'm starting complex analysis and I've learned about the required conditions for a complex valued function of complex variables to be derivable.

For the closest real analog, the $\mathbb R^2 \to\mathbb R^2$ function, has a differential which is a linear operator that best approximates the function at some point. This was a bit confusing at first. How can complex functions get away with having derivatives, again complex numbers, instead of linear operators.

Then I realized that the conditions given at the start actually demand that the differential is such that you can describe it's behavior with a single complex number and complex multiplication.

Now that I get that, I'm wondering what those operators look like. Geometrically, what's different about the way this kind of operator transforms a point?

Side note: By linear operator I mean a linear map. The naming differs in my language.

• You know how you can identify $\mathbb{C}$ with $\mathbb{R}^2$ as $\mathbb{R}$ vector spaces. Now look how a general $\mathbb{R}$-linear operator $\mathbb{R}^2 \to \mathbb{R}^2$ looks, and look how those specific ones that correspond to the multiplication with a complex number look like. – Daniel Fischer Feb 24 '15 at 13:58