proposition 17.96 Poles give asymptotics

open in the book · parts/02-mathematical-methods/15-fourier-integral-transforms.tex:3309 · p. 709

Rests on

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 ¶
    - 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 definition 7.45 Series ¶
      - depends_on theorem 7.8 Cauchy criterion ¶
      - 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 8.21 Laurent expansion ¶
  - 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 ¶
      - depends_on equation 8.5 eq:cpx-cr ¶
      - depends_on theorem 7.131 Green ¶
      - 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 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 ¶
depends_on lemma 8.9 ML estimate ¶ ↺
depends_on theorem 8.24 Residue theorem ¶
- depends_on corollary 8.15 Deformation of contours ¶ ↺
- depends_on definition 8.22 Isolated singularities; residue ¶ ↺
- 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 17.94 Mellin inversion ¶
- depends_on theorem 8.12 Cauchy ¶ ↺
- 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 ¶
- depends_on theorem 17.93 Fundamental strip and holomorphy ¶
  - depends_on definition 17.92 Mellin transform ¶
  - depends_on proposition 17.67 Holomorphy ¶
    - depends_on definition 8.5 Complex derivative ¶
    - depends_on definition 17.65 One-sided Laplace transform ¶
    - depends_on theorem 7.38 Taylor's theorem with Lagrange remainder ¶
      - depends_on proposition 7.30 Leibniz rule ¶
      - depends_on theorem 7.34 Rolle ¶
      - proves proof ch:05-real-analysis@proof-22 ¶
    - proves proof ch:15-fourier-integral-transforms@proof-40 ¶
  - proves proof ch:15-fourier-integral-transforms@proof-56 ¶
- proves proof ch:15-fourier-integral-transforms@proof-57 ¶
proves proof ch:15-fourier-integral-transforms@proof-58 ¶

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type	direction	node	provenance	where
`depends_on`	→	Isolated singularities; residue	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:3320
`depends_on`	→	ML estimate	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:3320
`depends_on`	→	Residue theorem	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:3320
`depends_on`	→	Mellin inversion	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:3320
`proves`	←	ch:15-fourier-integral-transforms@proof-58	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:3323