Complex Analysis¶
Job: holomorphic functions on \(\mathbb{C}\) — Cauchy theory, residues, and global constructions (Mittag-Leffler, Weierstrass) as used in analysis, number theory, and geometry.
Core¶
Holomorphicity is far more rigid than real differentiability: local power series, identity theorem, open mapping, and residue calculus follow from a single integral identity.
| Page | Quals focus |
|---|---|
| Foundations | \(\mathbb{C}\), CR equations, elementary maps, topology of domains |
| Analytic functions | holomorphicity \(\Leftrightarrow\) analyticity; identity; singularities |
| Complex integration | Cauchy, Goursat, homotopy invariance, estimates |
| Residue theory | residues, argument principle, Rouché; real integrals |
Theorem spine¶
- Goursat / Cauchy integral theorem on triangles \(\to\) primitive locally.
- Cauchy integral formula \(\to\) power series, infinite differentiability.
- Residue theorem \(\to\) contour integrals; argument principle \(\to\) zeros/poles counts.
- Mittag-Leffler / Weierstrass \(\to\) prescribed poles / zeros.
Interfaces¶
- Real analysis: several variables vs one complex
- Functional analysis: holomorphic functional calculus
- Algebraic topology: winding number \(= \deg\)
- Riemann surfaces / several complex variables: SCV, complex geometry
See it¶
Prime number theorem via \(\zeta(s)\): zero-free region + contour integration of \(-\zeta'/\zeta\) — complex analysis as engine of analytic number theory. Same package: Mellin transforms, Perron’s formula.
Check¶
Why does \(C^1\) in the real sense not imply holomorphic?
Holomorphic requires complex linearity of \(Df\) (Cauchy–Riemann). \(f(z)=\bar z\) is real \(C^\infty\) but nowhere complex differentiable.
Can a nonconstant entire function omit two values?
Picard: nonconstant entire omits at most one value (\(e^z\) omits \(0\)). Omitting two \(\Rightarrow\) constant (little Picard).
Pitfalls¶
- Domain connectivity matters for existence of global primitives and branches of \(\log\).
- Real-line intuition for maxima fails: nonconstant holom. has no interior max of \(|f|\) (maximum modulus).
- Branch cuts are artifacts of representing multi-valued inverses on \(\mathbb{C}\); Riemann surfaces remove them.
Refs¶
- Ahlfors, Complex Analysis
- Conway, Functions of One Complex Variable
- Stein–Shakarchi, Complex Analysis
Next¶
Foundations · Analytic functions · Complex integration · Residue theory · SCV · Complex geometry