Analytic Functions¶
Job: holomorphicity \(\Leftrightarrow\) local power series; identity principle; classification of isolated singularities; global existence theorems (Weierstrass, Mittag-Leffler).
Core¶
Definition. \(f\) is analytic on \(U\) if for each \(a\in U\) there is a power series \(\sum c_n(z-a)^n\) converging to \(f\) on some ball \(B(a,r)\subseteq U\).
Theorem (Holomorphic \(\Leftrightarrow\) analytic). On open \(U\subseteq\mathbb{C}\), complex differentiable on \(U\) \(\Leftrightarrow\) analytic on \(U\). Proof path: Cauchy integral formula \(\Rightarrow\) power series coefficients \(c_n=\frac{1}{2\pi i}\int\frac{f(\zeta)}{(\zeta-a)^{n+1}}d\zeta\).
Thus holomorphic \(\Rightarrow\) \(C^\infty\) and locally given by its Taylor series (contrast real \(C^\infty\)).
Consequences of analyticity¶
| Theorem | Statement |
|---|---|
| Identity theorem | \(f,g\) holo on domain \(U\); \(\{f=g\}\) has limit point in \(U\) \(\Rightarrow\) \(f\equiv g\) |
| Open mapping | nonconstant holo on domain \(\Rightarrow\) open map |
| Maximum modulus | \(\|f\|\) has no interior maximum unless \(f\) constant; min modulus if \(f\neq 0\) |
| Liouville | bounded entire \(\Rightarrow\) constant |
| FTA | nonconstant polynomial has a root (\(1/p\to 0\) at \(\infty\), else Liouville on \(1/p\)) |
| Schwarz lemma | \(\|f\|\le 1\) on \(\mathbb{D}\), \(f(0)=0\) \(\Rightarrow\) \(\|f(z)\|\le\|z\|\), \(\|f'(0)\|\le 1\); equality \(\Rightarrow\) rotation |
Isolated singularities¶
\(f\) holo on \(0<\|z-a\|<R\). Laurent series \(f(z)=\sum_{n=-\infty}^\infty c_n(z-a)^n\) on annuli.
| Type | Criterion | Laurent |
|---|---|---|
| Removable | \(\lim_{z\to a}f(z)\) exists finite (or bounded near \(a\)) | no negative terms |
| Pole order \(k\) | \((z-a)^k f\) holo nonzero at \(a\); \(k\) minimal | finite principal part, lowest \(-k\) |
| Essential | neither | infinitely many negative powers |
Casorati–Weierstrass: essential singularity \(\Rightarrow\) image of any punctured nbhd dense in \(\mathbb{C}\). Great Picard: omits at most one value infinitely often near essential singularity.
Riemann removable singularity theorem: bounded near isolated singularity \(\Rightarrow\) removable.
Zeros¶
Zeros of nonconstant holo \(f\) are isolated (unless \(f\equiv 0\)); finite order \(k\) with \(f(z)=(z-a)^k g\), \(g(a)\neq 0\) holo. Meromorphic: holo except poles.
Global constructions (one-liners)¶
Theorem (Weierstrass). On domain \(U\), prescribed discrete zeros with multiplicities \(\Rightarrow\) holo \(f\) with exactly those zeros (infinite products; for \(U=\mathbb{C}\), genus/Weierstrass factors \(E_n(z)\)).
Theorem (Mittag-Leffler). Prescribed principal parts at discrete poles \(\Rightarrow\) meromorphic \(f\) with those polar parts (partial fraction convergence tricks).
Corollary. Meromorphic on \(\widehat{\mathbb{C}}\) \(\Leftrightarrow\) rational. Entire of finite order controlled by Nevanlinna / Hadamard factorization.
See it¶
Gamma function. \(\Gamma(z)\) meromorphic on \(\mathbb{C}\): Weierstrass product \(\frac{1}{\Gamma(z)}=ze^{\gamma z}\prod_n(1+z/n)e^{-z/n}\); simple poles at nonpositive integers. Functional equation \(\Gamma(z+1)=z\Gamma(z)\) analytic continuation of factorial.
Elliptic functions. Doubly periodic meromorphic functions: Mittag-Leffler data constrained by residue sum \(0\) on period parallelogram; Weierstrass \(\wp\) has double poles on lattice.
Check¶
Prove Liouville from Cauchy estimates.
\(f\) entire, \(\|f\|\le M\). Cauchy: \(\|f^{(n)}(0)\|\le n! M/R^n\) all \(R\). Let \(R\to\infty\): \(f^{(n)}(0)=0\) for \(n\ge 1\); power series constant.
Counterexample in several real variables: why identity theorem fails for real analytic vs mere \(C^\infty\)?
Real analytic identity theorem holds on domains; \(C^\infty\) bump functions \(e^{-1/x^2}\) vanish on half-lines without vanishing identically. Complex holomorphicity is much stronger than real \(C^\infty\).
Pitfalls¶
- Radius of convergence of Taylor series may be smaller than distance to nearest singularity in a multi-valued continuation sense — for single-valued holo \(f\) on \(U\), radius equals distance to \(\partial U\) or nearest singularity of the germ’s continuation.
- Natural boundaries (lacunary series) block analytic continuation across arcs.
- Branch points are not isolated singularities of a single-valued function on a slit plane — they live on the Riemann surface.
Refs¶
- Ahlfors, Complex Analysis
- Conway, Functions of One Complex Variable
- Stein–Shakarchi, Complex Analysis
Next¶
Complex integration · Residue theory · Foundations · Complex analysis overview