All of complex analysis is filled with magical results like this. For example, Cauchy's formula and its generalizations. Max/Min results, Loiville's theorem, N

All of complex analysis is filled with magical results like this. For example, Cauchy's formula and its generalizations. Max/Min results, Loiville's theorem, Nevanlinna theory, Picard's Great and Little theorems. It's all absolutely bonkers. Just think: "Every complex function with an essential singularity achieves all but one value infinitely often in every neighborhood of the singularity" or "Every holomorphic function can be approximated arbitrarily well by a region of the Riemann Zeta function" There is this joke about how proofs start in different branches of math. So in real analysis you start with "Let ε > 0". And in Abstract Algebra, it's "Suppose not." But in Complex Analysis: "Nothing up my left sleeve, nothing up my right sleeve.."