theorem 17.100 Moments diagonalise a convolution evolution

open in the book · parts/02-mathematical-methods/15-fourier-integral-transforms.tex:3446 · p. 710

Rests on

depends_on corollary 17.99 Convolution on the unit interval ¶
- depends_on definition 17.92 Mellin transform ¶
- depends_on definition 17.97 Multiplicative convolution ¶
- depends_on theorem 17.98 Mellin convolution theorem ¶
  - depends_on definition 17.92 Mellin transform ¶ ↺
  - depends_on definition 17.97 Multiplicative convolution ¶ ↺
  - proves proof ch:15-fourier-integral-transforms@proof-59 ¶
- proves proof ch:15-fourier-integral-transforms@proof-60 ¶
depends_on theorem 17.94 Mellin inversion ¶
- 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 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-61 ¶

Supports

depends_on corollary 17.101 Coupled channels ¶

Neighborhood

Every logical edge within two steps of this node.

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partly declared
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Edges

type	direction	node	provenance	where
`depends_on`	→	Convolution on the unit interval	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:3466
`depends_on`	→	Mellin inversion	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:3466
`depends_on`	←	Coupled channels	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:3504
`proves`	←	ch:15-fourier-integral-transforms@proof-61	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:3469