Entire = holomorphic on ℂ. Liouville: bounded ⇒ constant. Little Picard: entire omits at most one value. Great Picard: essential singularity omits at most one.
graph TD
D1["Def: Entire function\nholomorphic on ℂ"]
D2["Def: Order of growth\nρ = lim sup log M(r)/log r"]
D3["Def: Essential sing\nLaurent: infinitely many\nnegative powers"]
T1["Thm: Liouville\nentire bounded ⇒ constant"]
T2["Thm: Little Picard\nentire omits ≥2 values\n⇒ constant"]
T3["Thm: Great Picard\nf has ess sing at z₀\n⇒ f omits at most 1\nin punctured nbhd"]
T4["Thm: Hadamard factorization\nentire order ρ"]
T5["Thm: Weierstrass product\nzeros ⇒ entire construction"]
L1["Lemma: Casorati–Weierstrass\nimage dense near ess sing"]
D1 --> T1
D1 --> T2
D2 --> T4
D3 --> T3
D3 --> L1
L1 --> T3
T2 --> T4
classDef definition fill:#b197fc,color:#fff
classDef theorem fill:#51cf66,color:#fff
classDef lemma fill:#74c0fc,color:#fff
class D1,D2,D3 definition
class T1,T2,T3,T4,T5 theorem
class L1 lemma
Process Statistics
- Nodes: 15
- Edges: 11
- Definitions: 3
- Theorems: 5
- Lemmas: 1
Frontier: Nevanlinna theory, value distribution, complex dynamics (iteration of entire maps). math.CV