proposition 17.77 Poles and stability

open in the book · parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2735 · p. 702

Rests on

depends_on corollary 17.71 Residue evaluation and causality ¶
- depends_on lemma 17.70 Jordan's lemma on the Bromwich contour ¶
  - 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:15-fourier-integral-transforms@proof-43 ¶
- 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 ¶
- depends_on theorem 8.24 Residue theorem ¶
  - 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 6.15 Path-connected space ¶
    - depends_on theorem 8.12 Cauchy ¶ ↺
    - proves proof ch:06-complex-analysis@proof-9 ¶
  - depends_on definition 8.22 Isolated singularities; residue ¶
    - 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 ¶
      - proves proof ch:06-complex-analysis@proof-6 ¶
    - depends_on theorem 8.21 Laurent expansion ¶
      - depends_on corollary 8.15 Deformation of contours ¶ ↺
      - depends_on lemma 8.11 The fundamental $2\pi\ii$ ¶ ↺
      - depends_on theorem 8.16 Cauchy integral formula ¶
      - depends_on theorem 8.20 Taylor expansion ¶
      - proves proof ch:06-complex-analysis@proof-15 ¶
  - depends_on lemma 8.11 The fundamental $2\pi\ii$ ¶ ↺
  - depends_on theorem 8.21 Laurent expansion ¶ ↺
  - proves proof ch:06-complex-analysis@proof-17 ¶
- depends_on theorem 17.69 Bromwich inversion integral ¶
  - depends_on proposition 17.68 The Laplace transform is a Fourier transform ¶
    - depends_on definition 17.65 One-sided Laplace transform ¶
    - depends_on definition 17.29 Fourier transform; the treatise convention ¶
    - proves proof ch:15-fourier-integral-transforms@proof-41 ¶
  - depends_on theorem 17.32 Fourier inversion ¶
    - depends_on definition 17.29 Fourier transform; the treatise convention ¶ ↺
    - depends_on equation 17.42 eq:ft-gaussian ¶
    - proves proof ch:15-fourier-integral-transforms@proof-20 ¶
  - proves proof ch:15-fourier-integral-transforms@proof-42 ¶
- proves proof ch:15-fourier-integral-transforms@proof-44 ¶
proves proof ch:15-fourier-integral-transforms@proof-48 ¶

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type	direction	node	provenance	where
`depends_on`	→	Residue evaluation and causality	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2742
`proves`	←	ch:15-fourier-integral-transforms@proof-48	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2745