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Complex analysis is far too nice. There are so many theorems where my first thought was "there's no way that's true". Picard's great theorem is totally insane.

• lanstin 2y ago
  And the definite integrals you can get out of the Cauchy integral formula are awesome. I think it is because being holomorphic is so much stronger even than C^\infty (continuous with all derivatives continuous) real much less just continuous or merely integrable.
• btilly 2y ago
  Right. It's magic. But once you have the machine... ----- Fundamental Theorem of Algebra: Every nonconstant polynomial over the complex numbers has a complex root. Proof: Suppose that p(z) is a polynomial over the complex numbers with no root. Consider the function 1/p(z). If p(z) had no roots, then 1/p(z) is entire. But we can bound it for everything outside of a large circle because the leading term dominates the others. And since the large circle is compact, we can bound p(z) away from 0 inside the circle. Between the two, 1/p(z) is bounded, and so much be constant by Liouville's theorem. But 1/p(z) is only constant if p(z) is constant. Therefore any complex polynomial with no complex roots must be constant. ----- I'm convinced by the proof. But part of me still says that it is magic.