Complex

October 2023.

Something I vaguely remember from undergrad is that Cauchy developed complex analysis from an analytic point of view, and then Riemann came along and re-did everything from a geometric and abstraction function theory point of view. Despite some efforts from Weierstrass, Riemann’s view dominated, and today complex analysis is mainly taught that way.

But: The analytic perspective is under-rated. If you’re willing to hand-wave convergence details (or work them out once), the analytic perspective makes it way easier to remember the basic theorems and to have some intutition on how to prove them. For example, let’s assume that

$$f(z) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)z^n}{n!}$$

on some open disk $D$ around the origin, and that the convergence is uniform. Then:

1. $f(z)$ is infinitely differentiable. Proof: Trivial (differentiate terms).
2. Closed curve theorem: $\int_{\gamma}f(z) = 0$ for any closed curve $\gamma$ in $D$. Proof: Clearly $f$ has an anti-derivative, $F(z) = \sum f^{(n)}(0)z^{n+1}/(n+1)!$.
3. Cauchy integral formula: Let $\gamma$ be a circle around the origin with $\gamma \subset D$. Then $\int_{\gamma}f(z)/z^{n+1} = f^{(n)}(0)2\pi i / n!$. Proof: Use the power series for $f(z)/z^n$. All the terms have anti-derivatives except for $f^{(n)}(0)/(n!z)$, which integrates to the desired result. You only have to prove the theorem for $1/z$.
4. Schwarz reflection principle. Proof: Two power series that agree on any set with an accumulation point must have equal coefficients.
5. Etc., etc.

The point is that if you can identify a holomorphic function with its power series (and you basically can!), then all the elemntary theorems are trivial. I find it much easier to remember things this way. As Elias Stein said, it’s a lot easier to prove something rigorously once you know the “real” reason why it’s true.