lemma 8.11 The fundamental $2\pi\ii$

open in the book · parts/02-mathematical-methods/06-complex-analysis.tex:227 · p. 267

Rests on

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 ¶
      - depends_on definition A.36 Cut ¶
      - depends_on definition A.38 Order on $\R$ ¶
      - proves proof app:A-long-proofs@proof-29 ¶
    - 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 definition 7.2 Absolute value ¶ ↺
  - depends_on theorem 7.8 Cauchy criterion ¶
    - depends_on corollary A.47 Cauchy completeness ¶
      - depends_on corollary A.46 Monotone convergence ¶ ↺
      - depends_on theorem A.40 Least-upper-bound property ¶ ↺
      - proves proof app:A-long-proofs@proof-34 ¶
    - 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 ¶

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 8.16 Cauchy integral formula ¶
- depends_on theorem 8.17 Derivatives of all orders; Cauchy estimates ¶
  - depends_on theorem 8.18 Liouville ¶
    - depends_on corollary A.723 Potential of one infinite row ¶
      - depends_on definition A.724 The staggered street ¶
      - depends_on proposition A.725 The street translates rigidly ¶
    - depends_on lemma A.608 An entire function with a quadratic bound on its real part ¶
      - depends_on proposition A.609 The exponent is a quadratic polynomial ¶
    - depends_on theorem 8.19 Fundamental theorem of algebra ¶
      - depends_on corollary 5.72 Existence of an eigenvalue over $\C$ ¶
      - depends_on lemma 15.12 Artin decomposition ¶
      - depends_on proposition 9.28 The scalar constant-coefficient equation ¶
    - depends_on theorem 12.54 The spectrum is compact and non-empty ¶
      - depends_on theorem 12.55 The spectrum of a self-adjoint operator is real ¶
- depends_on theorem 8.21 Laurent expansion ¶
  - depends_on definition 8.22 Isolated singularities; residue ¶ ↺
  - depends_on theorem 8.24 Residue theorem ¶ ↺
depends_on theorem 8.21 Laurent expansion ¶ ↺
depends_on theorem 8.24 Residue theorem ¶ ↺

Neighborhood

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partly declared
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Edges

type	direction	node	provenance	where
`depends_on`	→	eq:cpx-contour-integral	declared	parts/02-mathematical-methods/06-complex-analysis.tex:237
`depends_on`	→	Euler's formula	declared	parts/02-mathematical-methods/06-complex-analysis.tex:237
`depends_on`	←	Isolated singularities; residue	declared	parts/02-mathematical-methods/06-complex-analysis.tex:580
`depends_on`	←	Cauchy integral formula	declared	parts/02-mathematical-methods/06-complex-analysis.tex:409
`depends_on`	←	Laurent expansion	declared	parts/02-mathematical-methods/06-complex-analysis.tex:539
`depends_on`	←	Residue theorem	declared	parts/02-mathematical-methods/06-complex-analysis.tex:610
`proves`	←	ch:06-complex-analysis@proof-6	declared	parts/02-mathematical-methods/06-complex-analysis.tex:240