I am new to analysis and just starting to appreciate it. So far I have encountered 3 what seem like fundamental links between the cases of $\mathbb{R}$ and $\mathbb{C}$:
- The Cauchy-Riemann equations
- The Paley-Wiener theorem
- The fact that the open disk of convergence of a power series is ${\it exactly}$ the maximal one on which it can be extended holomorphically (no smaller). In particular this gives us a "geometric explanation'' for the radii of convergence of the Taylor series for, say, $\frac{1}{1 + z^2}$, about each $\alpha \in \mathbb{R}$.
Question: can we view these connections as all arising from one? Are there any fundamentally "different" connections?