I remember doing a taylor series on that one, where e^z = 1 +z + z^2 /2 + z^3/3! + ...
I can't assume anything about complex differentiability yet
At the time I wanted to avoid it because of the fact we hadn't done differentiation but turns out it was the next lecture lol. So I ended up just L'H smashing it.I remember doing a taylor series on that one, where e^z = 1 +z + z^2 /2 + z^3/3! + ...
How does that make it any less cheating?At the time I wanted to avoid it because of the fact we hadn't done differentiation but turns out it was the next lecture lol. So I ended up just L'H smashing it.
Which is probably me cheating, because this is the definition of the derivative, but I'm in C so I'll cheat by using the Cauchy-Riemann equations instead
That being said,How does that make it any less cheating?
You should use the definition of the exponential explicitly (presumably it is the power series centred at 0, so the argument is much the same). Don't assume anything about the differentiability of the exponential if the limit you are asked to evaluate (if it exists) is the definition of the derivative.
Same with L'Hopital. Eg you could try to find the limit of sin(x)/x as x tends to zero by using L'Hopital and it "works" in that cos(x)/1 clearly does tend to 1. However how did we prove that sin(x) has derivative cos(x)? It turns out we need to essentially evaluate the limit of sin(x)/x in order to do this!
Because working with power series is pretty low level analysis, you don't need much theory development before you can start saying meaningful things about functions defined via power series. (Eg it is smooth and invariant under differentiation, which almost uniquely characterises the exponential.)That being said,
When you first told me that the power series was the definition of the exponential I initially accepted it, but became reluctant to believe. Why is that the case? Or do they just teach things in the wrong order?
Because the power series falls out of differentiation as well.
I get your point, but I really don't know how I feel about the power series definition. I would've thought that the limit definition was more acceptable. (Hopefully I got that right because I didn't double check)