Entire and Meromorphic Functions

1799, Göttingen. The 22-year-old Carl Friedrich Gauss proved the Fundamental Theorem of Algebra geometrically in his doctoral thesis. By 1816 he had three more proofs in hand; a fourth came in 1849. But the shortest and most elegant proof appeared with Joseph Liouville in 1844: a bounded entire function must be constant, so $1/p(z)$ for a polynomial without zeros leads to a contradiction. Today the same rigidity of entire functions underlies the Nyquist-Shannon sampling theorem: a band-limited signal is an entire function of finite exponential type, hence uniquely recoverable from discrete samples. One class of functions, from Gauss to 5G.

• **Nyquist-Shannon sampling theorem:** a band-limited signal $f(t)$ is entire of exponential type $\sigma$; by Paley-Wiener it is uniquely reconstructed from samples at rate $2\sigma$ - the foundation of all digital communication
• **Riemann zeta function:** $\zeta(s)$ is meromorphic on $\mathbb{C}$ with a single simple pole at $s = 1$; the location of its zeros is the content of the 1859 Riemann hypothesis
• **Euler gamma function:** $\Gamma(s)$ is meromorphic with poles at $s = 0, -1, -2, \ldots$, and the Weierstrass factorization gives $1/\Gamma(s) = s e^{\gamma s} \prod (1 + s/n) e^{-s/n}$
• **Euler product for $\sin$:** $\sin(\pi z) = \pi z \prod_{n \geq 1} (1 - z^2/n^2)$ - the first historical instance of Weierstrass factorization (Euler, 1734)

Предварительные знания

• Möbius Transformations

Entire Functions and Liouville's Theorem

An entire function is analytic on all of C. Liouville's theorem reveals the extreme rigidity of complex analysis: a bounded entire function must be constant - something with no real-analysis analogue.

**Liouville's Theorem:** a bounded entire function is constant. **Proof sketch:** Cauchy's inequality gives |f'(z)| ≤ M/R for any R. Let R→∞ → f'(z)=0 → f=const. **Polynomial Liouville:** if |f(z)| ≤ C(1+|z|)ⁿ for all z, then f is a polynomial of degree ≤ n. **Growth order:** f has order ρ if |f(z)| ≤ e^{|z|^ρ} for large |z|. - eᶻ: order 1 - e^{z²}: order 2 - Polynomials: order 0

The Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra - every nonconstant polynomial has a complex root - follows almost instantly from Liouville. The complex-analysis proof is one of the shortest known.

**Fundamental Theorem of Algebra:** every polynomial p(z) of degree n ≥ 1 has at least one root in C. **Proof via Liouville:** 1. Suppose p(z) ≠ 0 for all z. 2. Then 1/p(z) is entire. 3. Since |p(z)| → ∞ as |z|→∞, |1/p(z)| → 0 → 1/p is bounded. 4. Liouville: 1/p = const → contradiction. **Corollary:** p(z) factors completely over C: p(z) = aₙ(z−r₁)···(z−rₙ)

Weierstrass Factorization Theorem

The Weierstrass Factorization Theorem generalizes the polynomial factorization p(z) = a·Π(z−rₖ) to entire functions. Any entire function can be expressed as a product over its zeros, times an exponential factor.

**Weierstrass Factorization:** any entire function with zeros {aₙ} (repeated by multiplicity) is: f(z) = zᵐ · e^{g(z)} · Πₙ Eₚₙ(z/aₙ) where Eₚ(z) = (1−z)·exp(z + z²/2 + ... + zᵖ/p) are elementary factors. **Hadamard's theorem:** if |f(z)| ≤ C·e^{|z|^ρ}, then: - g is a polynomial of degree ≤ ρ - The product converges if Σ|aₙ|^{−ρ−1} < ∞ **Classic example:** sin(πz) = πz · Πₙ₌₁^∞ (1−z²/n²)

Picard's Theorems

Picard's theorems describe how "densely" an entire or meromorphic function covers C. The results are striking: a non-polynomial entire function misses at most one value.

**Little Picard Theorem:** a nonconstant entire function takes every value in C, with at most one exception. Example: eᶻ ≠ 0 but takes every other value. **Great Picard Theorem:** in any punctured neighborhood of an essential singularity, f takes every value in C, with at most one exception. Example: e^{1/z} near z=0: takes every value except 0. **Contrast with poles:** near a pole, |f(z)| → ∞ (misses all small values). Essential singularities are far more pathological.

Key Ideas

• **Liouville:** bounded entire = constant; proof via Cauchy inequality as R→∞
• **FTA:** every polynomial has a root in C - proved in two lines via Liouville
• **Weierstrass:** f = zᵐ·e^{g}·Πₙ Eₚₙ(z/aₙ) - factors entire functions over their zeros
• **Picard:** entire function misses at most one value; essential singularity misses at most one near z₀

Connected Topics

Entire functions connect algebra, analysis, and number theory:

• Analytic Continuation — The Riemann zeta function - a meromorphic continuation - uses these results as foundation
• Harmonic Functions — Real and imaginary parts of entire functions are harmonic - the bridge to PDE theory
• Complex Analysis in CS — Entire functions of finite order appear as generating functions in algorithm analysis

Вопросы для размышления

• Why does Liouville's theorem fail for real-analytic functions? Give a counterexample.
• How does Weierstrass factorization explain why sin has Euler's infinite product formula?
• The Great Picard Theorem says e^{1/z} takes all values except 0 near z=0. How can you visualize this geometrically using the Riemann sphere?

Связанные уроки

• calc-15-convergence