To quote Mr. Cartman, "Fuck fuckety fuck fuck fuck." This is really terribly frustrating. What, you may ask yourself, has reduced me to quoting South Park? It's not still not knowing how to prove that there's an M(z) taking two non-concentric non-intersecting circles to concentric circles. (Though I still haven't solved that.) It's not having a Tom Petty song stuck in my head. (That's actually the case, but I don't mind.) And it's not even the (relatively stale by now) news that Steven Jay Gould died. (Though that's definetly a downer.)

No, indeedy. See, as talked about in the last entry, I was struggling with a particular problem on Mobius transforms. I've made no headway. So I thought I'd go on ahead. And there are some neat problems that crop up toward the end of that chapter, let me tell you. Problem is, I've found that I'm temporarily sick of M(z)s. So I decided to just read the next chapter. This was a good decision, as it turns out, because just as I remember, this is a fun chapter - basically, 'differentiation of complex functions' (I'd read it before).

So, you ask, what's neat about it? Well, for instance, there's the concept of the derivative as an amplitwist. What this means is that if you take the derivative of a complex function at a point (f'(z)), you get a complex number. Now, what does that number *mean*? Well, multiplication of something by a complex number (r*e^{i*theta}) means you're expanding that something (that's the 'r' part), and rotating it (that's the e^{i*theta}). Another way of saying that is that you're 'amplifying and twisting', or amplitwisting, in short. So, that complex number that you get from taking the derivative tells you how you need to amplitwist an infinitesmal vector poking out from z to get the 'image vector' poking out from f(z). So, if instead of thinking about points, let's think of tiny little figures, triangles, say. Now the 'amplitwist' tells us how a teeny weeny little triangle around z is going to twisted and amplified by the transformation we've just differentiated. As expected, it tells us how an infinitesemal chunk of a the transform behaves.

Along the way in figuring all this out, there's a brilliantly simple derivation of the Jacobian, and an extremely illuminating derivation of the Cauchy-Riemann equations. (Basically, take the concept of the amplitwist, combine it with the concept of the Jacobian, and the Cauchy-Riemann equations pop right out!). There is other good stuff, too.

But what got me peeved is the following. The derivative of a constant better be zero, right? Right. It's the same in complex analysis. So, back when I was reading the chapter, it had this statement, and warned that I must make sure to see it, and that it isn't obvious. So I sat, and huffed, and puffed, and concluded that I did see it, and while not entirely obvious, it was really quite simple, and even obvious-like in a crafty way. Then I moved on. Now I'm doing the associated exercises. And lo and behold, I'm asked to show why the derivative of a constant is zero. And I've forgotten how to do this! What's worse is that I KNEW how to do it. I just forgot. And I can't seem to remember.

"Fuck fuckety fuck fuck fuck"