∫_γ f(z) dz = 0 for null-homologous γ. Cauchy integral formula, residue theorem. Main computational tool for contour integrals.
graph TD
D1["Def: Contour integral\n∫_γ f dz"]
D2["Def: Winding number\nn(γ,z)"]
D3["Def: Residue\nRes(f,z₀)"]
T1["Thm: Cauchy–Goursat\n∫_γ f=0 if f holomorphic\ninside γ"]
T2["Thm: Cauchy formula\nf(z)=1/2πi ∫ f(ζ)/(ζ−z) dζ"]
T3["Thm: Residue theorem\n∫_γ f = 2πi Σ Res(f,zⱼ)"]
T4["Thm: Argument principle\n∫ f'/f = 2πi(N−P)"]
T5["Thm: Rouché\n|f−g|<|g| ⇒ same # zeros"]
L1["Lemma: Laurent coeff\nc₋₁ = Res"]
T1 --> T2
T2 --> T3
T2 --> L1
L1 --> T3
T3 --> T4
T4 --> T5
D1 --> T1
D2 --> T4
D3 --> T3
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: 14
- Definitions: 3
- Theorems: 5
- Lemmas: 1
Frontier: Tropical residues, higher residues, applications to mirror symmetry. math.CV