Complex Analysis
These terms actually mean something
Analytic - https://en.wikipedia.org/wiki/Analytic_function. A function is analytic on an open set if a convergent power series exists on that
Holomorphic - A function is holomorphic on an open set if it is complex differentiable on every point of the set/
That these two are the same for complex variables is remarkable and deserving of proof.
In analysis, the proof of limits being defined, convergence, etc, involves bounds. Carrying along proofs has been too burdensome
What about trying to package this info alongside functions
Riemann Surfaces
Uniformization theorem every simply connected riemann surface is conformally equivalent to an open disk, plane, or riemann sphere
Convergence
https://en.wikipedia.org/wiki/Convergent_series https://en.wikipedia.org/wiki/Cauchy_sequence
Cauchy sequences get closer and closer together
(nat -> R, eps -> N)
We can add and multiply convergent sequences. Can I take the partial sum over a convergent sequence to get a new convergent sequence? No. Of course not. I can finite difference a sequence, shift, I can’t invert.
Composition doesn’t even really make sense.
uniformly continuous functions can be applied to
There is a lattice of properties (I think). Differentiable implies continuity and so on.
Continuity
Contour Integrals
Misc
Circle packing
Books:
- Visual Complex Analysis
- Gamelin
-
Ahlfors