proposition 8.4 Euler's formula

open in the book · parts/02-mathematical-methods/06-complex-analysis.tex:80 · p. 266

Rests on

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 $\R$ ¶
      - depends_on definition A.36 Cut ¶ ↺
    - 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 ¶

Supports

depends_on lemma A.786 From the half-strip to the half plane ¶
- depends_on corollary A.788 The two parameters are the same parameter ¶
  - depends_on theorem A.789 Kirchhoff's contraction coefficient ¶
- depends_on lemma A.787 From the potential strip to the half plane ¶
  - depends_on corollary A.788 The two parameters are the same parameter ¶ ↺
depends_on lemma 8.11 The fundamental $2\pi\ii$ ¶
- 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 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 lemma A.608 An entire function with a quadratic bound on its real part ¶
      - depends_on theorem 8.19 Fundamental theorem of algebra ¶
      - depends_on theorem 12.54 The spectrum is compact and non-empty ¶
  - 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 ¶ ↺
depends_on lemma 17.14 Uniformly bounded sine sums ¶
- depends_on lemma 17.15 A polynomial whose partial sum spikes at the origin ¶
  - depends_on proposition 17.16 An explicit continuous function with a divergent Fourier series ¶

Neighborhood

Every logical edge within two steps of this node.

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partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	eq:cpx-exp-series	declared	parts/02-mathematical-methods/06-complex-analysis.tex:87
`depends_on`	→	Comparison; absolute convergence	declared	parts/02-mathematical-methods/06-complex-analysis.tex:87
`depends_on`	←	From the half-strip to the half plane	declared	appendices/A-long-proofs.tex:38603
`depends_on`	←	The fundamental $2\pi\ii$	declared	parts/02-mathematical-methods/06-complex-analysis.tex:237
`depends_on`	←	Uniformly bounded sine sums	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:593
`proves`	←	ch:06-complex-analysis@proof-2	declared	parts/02-mathematical-methods/06-complex-analysis.tex:90