Holomorphic f(z)=u+iv: complex differentiability, Cauchy–Riemann conditions uₓ=vᵧ, uᵧ=−vₓ. Equivalent to complex analytic (power series).
graph TD
D1["Def: ℂ as ℝ²\nz = x + iy"]
D2["Def: Complex derivative\nf'(z)=lim f(z+h)−f(z)/h"]
D3["Def: Holomorphic\nf' exists on Ω"]
D4["Def: u,v real/imag\nf = u + iv"]
T1["Thm: Cauchy–Riemann\nuₓ=vᵧ, uᵧ=−vₓ\n⇔ complex diff"]
T2["Thm: Holomorphic ⇒\nu,v harmonic Δu=Δv=0"]
T3["Thm: Equivalence\nholomorphic ⇔ analytic\n(power series)"]
T4["Thm: Open mapping\nf nonconst ⇒ f open"]
T5["Thm: Identity principle\nzeros isolated or f≡0"]
L1["Lemma: Wirtinger\nd/dz̄ = 0"]
D2 --> D3
D4 --> T1
D3 --> T1
T1 --> T2
T1 --> L1
T3 --> T4
T3 --> T5
classDef definition fill:#b197fc,color:#fff
classDef theorem fill:#51cf66,color:#fff
classDef lemma fill:#74c0fc,color:#fff
class D1,D2,D3,D4 definition
class T1,T2,T3,T4,T5 theorem
class L1 lemma
Process Statistics
- Nodes: 16
- Edges: 13
- Definitions: 4
- Theorems: 5
- Lemmas: 1
Frontier: Several complex variables, CR manifolds, PDE connections. math.CV