Proofs of the fundamental theorem of algebra, by the argument principle, Rouché’s theorem, Liouville’s theorem, and the open mapping theorem.
The Gamma function and conic sections, the \(\Gamma\)-function identities and the standard conics.
The Gauss--Lucas theorem — the zeros of \(p'\) lie in the convex hull of the zeros of a polynomial \(p\).
The Dirichlet problem, solved with real and imaginary parts of holomorphic functions.
Special functions — Fourier transforms, \(\Gamma\), \(\zeta\), and elliptic functions.