theorem 8.21 Laurent expansion

open in the book · parts/02-mathematical-methods/06-complex-analysis.tex:531 · p. 271

Rests on

depends_on corollary 8.15 Deformation of contours ¶
- depends_on definition 6.17 Simply connected space ¶
  - depends_on definition 6.6 Continuous map ¶
    - depends_on definition 3.53 Preimage ¶
    - depends_on definition 6.2 Open set ¶
      - depends_on definition 6.1 Topological space ¶
    - depends_on definition 6.1 Topological space ¶ ↺
  - depends_on definition 6.15 Path-connected space ¶
    - depends_on definition 6.6 Continuous map ¶ ↺
- depends_on theorem 8.12 Cauchy ¶
  - depends_on equation 8.5 eq:cpx-cr ¶
  - depends_on theorem 7.131 Green ¶
    - depends_on definition 7.127 Simple regions ¶
      - depends_on definition 7.98 Functions of class $C^{1}$ ¶
      - depends_on definition 6.9 Compact set ¶
    - depends_on remark 7.128 What the derivations below take as given ¶
      - depends_on definition 7.125 Multiple integral ¶
      - depends_on theorem 7.40 Continuous functions are integrable ¶
      - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
    - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
      - depends_on definition 7.41 Antiderivative ¶
      - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
      - proves proof ch:05-real-analysis@proof-25 ¶
    - proves proof ch:05-real-analysis@proof-77 ¶
  - proves proof ch:06-complex-analysis@proof-7 ¶
- proves proof ch:06-complex-analysis@proof-9 ¶
depends_on lemma 8.11 The fundamental $2\pi\ii$ ¶
- depends_on equation 8.6 eq:cpx-contour-integral ¶
- depends_on proposition 8.4 Euler's formula ¶
  - depends_on equation 8.2 eq:cpx-exp-series ¶
  - depends_on proposition 7.47 Comparison; absolute convergence ¶
    - depends_on corollary A.46 Monotone convergence ¶
      - depends_on theorem A.40 Least-upper-bound property ¶
      - proves proof app:A-long-proofs@proof-33 ¶
    - depends_on definition 7.45 Series ¶
      - depends_on definition 7.2 Absolute value ¶
      - depends_on definition 7.4 Convergence ¶
    - depends_on theorem 7.8 Cauchy criterion ¶
      - depends_on corollary A.47 Cauchy completeness ¶
      - depends_on definition 7.4 Convergence ¶ ↺
      - proves proof ch:05-real-analysis@proof-5 ¶
    - proves proof ch:05-real-analysis@proof-28 ¶
  - proves proof ch:06-complex-analysis@proof-2 ¶
- proves proof ch:06-complex-analysis@proof-6 ¶
depends_on theorem 8.16 Cauchy integral formula ¶
- depends_on corollary 8.15 Deformation of contours ¶ ↺
- depends_on lemma 8.11 The fundamental $2\pi\ii$ ¶ ↺
- depends_on lemma 8.9 ML estimate ¶
  - depends_on equation 8.6 eq:cpx-contour-integral ¶ ↺
  - proves proof ch:06-complex-analysis@proof-4 ¶
- proves proof ch:06-complex-analysis@proof-10 ¶
depends_on theorem 8.20 Taylor expansion ¶
- depends_on equation 8.12 eq:cpx-cif ¶
- depends_on equation 8.13 eq:cpx-derivative-formula ¶
- depends_on lemma 8.9 ML estimate ¶ ↺
- depends_on proposition 7.46 Geometric series ¶
  - depends_on corollary A.46 Monotone convergence ¶ ↺
  - depends_on definition 7.45 Series ¶ ↺
  - proves proof ch:05-real-analysis@proof-27 ¶
- proves proof ch:06-complex-analysis@proof-14 ¶
proves proof ch:06-complex-analysis@proof-15 ¶

Supports

depends_on definition 8.22 Isolated singularities; residue ¶
- depends_on proposition 8.23 Residue at a simple pole ¶
- depends_on proposition 17.96 Poles give asymptotics ¶
- depends_on theorem 8.24 Residue theorem ¶
  - depends_on corollary 17.71 Residue evaluation and causality ¶
    - depends_on corollary 17.76 Heaviside's expansion theorem ¶
    - depends_on proposition 17.77 Poles and stability ¶
  - depends_on example 8.25 A real integral by residues ¶
  - depends_on proposition 17.96 Poles give asymptotics ¶ ↺
  - depends_on proposition 17.61 Mean-square response of a damped resonator ¶
  - depends_on theorem 31.68 Kutta–Joukowski ¶
    - depends_on proposition 31.70 The lift-curve slope of a thin aerofoil ¶
    - depends_on remark 31.69 The Kutta condition, and where the circulation comes from ¶
    - depends_on theorem A.737 The lift-curve slope of the flat plate ¶
  - depends_on theorem 9.26 Structure of the solutions ¶
    - depends_on corollary 9.27 Linear stability ¶
      - depends_on corollary 9.41 Hill's equation ¶
      - depends_on proposition 9.31 Lyapunov's first method: asymptotic stability ¶
      - depends_on proposition 9.29 The planar classification ¶
      - depends_on theorem 9.39 Floquet ¶
depends_on theorem 8.24 Residue theorem ¶ ↺

Neighborhood

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declared and complete
partly declared
a check failed
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Edges

type	direction	node	provenance	where
`depends_on`	→	Deformation of contours	declared	parts/02-mathematical-methods/06-complex-analysis.tex:539
`depends_on`	→	The fundamental $2\pi\ii$	declared	parts/02-mathematical-methods/06-complex-analysis.tex:539
`depends_on`	→	Cauchy integral formula	declared	parts/02-mathematical-methods/06-complex-analysis.tex:539
`depends_on`	→	Taylor expansion	declared	parts/02-mathematical-methods/06-complex-analysis.tex:539
`depends_on`	←	Isolated singularities; residue	declared	parts/02-mathematical-methods/06-complex-analysis.tex:580
`depends_on`	←	Residue theorem	declared	parts/02-mathematical-methods/06-complex-analysis.tex:610
`proves`	←	ch:06-complex-analysis@proof-15	declared	parts/02-mathematical-methods/06-complex-analysis.tex:542