Monday, November 14th, 2011 | Author: Konrad Voelkel
Here is my mind-map for first-course complex analysis. It contains some well-known theorems and "arrows" between them.
There are some aspects which require an explanation:
An arrow (A --> B) roughly means "you could try to prove B by using A and maybe something else, although A might not be necessary to prove B". I tried to remove redundant arrows, so keep in mind you can prove most of the theorems from the Cauchy-Riemann differential equations and some calculus ...
Another hint on the interpretation of my mind-map:
"existence of anti-derivatives" in the context of the mind-map really means "holomorphic functions have anti-derivatives on every simply connected domain", and "homotopy-invariance of path integral" really means "the path integral over holomorphic functions is homotopy-invariant on every simply connected domain". While you can do complex analysis without homotopy theory (even without homology theory), I don't like that approach very much. At least by using homology, some statement are easier to state. With homotopy, they become easier to prove (assuming you are allowed to use algebraic topology).
I created the mind-map (on the occasion of my oral exams) to keep it as a reminder and to sort out which statements could be generalized to harmonic maps or even to analytic maps (that was fun!).
An incomplete list of theorems contained:
- Goursat's Lemma
- Cauchy's integral formula
- Liouville's Theorem
- Maximum Modulus Principle
- Open Mapping Theorem
- Identity Theorem
- Riemann's Removable Singularity Theorem
- Residue Theorem
- Fundamental Theorem of Algebra
- Montel's Theorem
- Schwarz's Lemma
- Riemann Mapping Theorem
It's a nice exercise to try to prove these on your own.
Enjoy learning, repeating or staring at confusing arrows!