proposition 17.67 Holomorphy

open in the book · parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2438 · p. 699

Rests on

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 definition 7.12 Product of functions ¶
    - depends_on definition 7.9 Real function ¶
      - depends_on definition 3.43 Map ¶
  - depends_on equation 7.14 eq:ana-derivh ¶
  - depends_on proposition 7.27 Differentiable implies continuous ¶
    - depends_on definition 7.20 Continuity at a point ¶
      - depends_on definition 7.16 Limit ¶
      - depends_on equation 7.7 eq:ana-limit-left ¶
      - depends_on equation 7.5 eq:ana-limit-right ¶
    - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on definition 7.16 Limit ¶ ↺
      - depends_on definition 7.9 Real function ¶ ↺
    - depends_on proposition 7.6 Algebra of limits ¶
      - depends_on definition 7.4 Convergence ¶
      - depends_on proposition 7.3 Triangle inequality ¶
      - proves proof ch:05-real-analysis@proof-3 ¶
    - proves proof ch:05-real-analysis@proof-11 ¶
  - depends_on proposition 7.6 Algebra of limits ¶ ↺
  - proves proof ch:05-real-analysis@proof-14 ¶
- depends_on theorem 7.34 Rolle ¶
  - depends_on lemma 7.33 Fermat: interior extremum ¶
    - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
    - depends_on proposition 7.18 Two-sided limit from one-sided limits ¶
      - depends_on definition 7.16 Limit ¶ ↺
      - depends_on equation 7.7 eq:ana-limit-left ¶ ↺
      - depends_on equation 7.5 eq:ana-limit-right ¶ ↺
      - proves proof ch:05-real-analysis@proof-6 ¶
    - proves proof ch:05-real-analysis@proof-17 ¶
  - depends_on theorem 7.24 Extreme value theorem ¶
    - depends_on axiom 7.1 Completeness of $\R$ ¶
    - depends_on proposition 7.22 Sequential characterization ¶
      - depends_on definition 7.20 Continuity at a point ¶ ↺
      - depends_on definition 7.4 Convergence ¶ ↺
      - proves proof ch:05-real-analysis@proof-7 ¶
    - depends_on theorem 7.7 Bolzano–Weierstrass ¶
      - depends_on corollary A.45 Archimedean property and density of $\Q$ ¶
      - depends_on corollary A.46 Monotone convergence ¶
      - proves proof ch:05-real-analysis@proof-4 ¶
    - proves proof ch:05-real-analysis@proof-9 ¶
  - proves proof ch:05-real-analysis@proof-18 ¶
- proves proof ch:05-real-analysis@proof-22 ¶
proves proof ch:15-fourier-integral-transforms@proof-40 ¶

Supports

depends_on theorem 17.78 Causality implies dispersion relations ¶
- depends_on remark 33.10 Fibre anelasticity, and why it biases one method and not the other ¶
- depends_on remark 28.26 Causality is visible in the susceptibility ¶
- depends_on remark 28.62 Neither velocity is a speed limit ¶
depends_on theorem 17.93 Fundamental strip and holomorphy ¶
- depends_on theorem 17.94 Mellin inversion ¶
  - depends_on proposition 17.96 Poles give asymptotics ¶
  - depends_on theorem 17.100 Moments diagonalise a convolution evolution ¶
    - depends_on corollary 17.101 Coupled channels ¶

Neighborhood

Every logical edge within two steps of this node.

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

Edges

type	direction	node	provenance	where
`depends_on`	→	Complex derivative	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2445
`depends_on`	→	One-sided Laplace transform	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2445
`depends_on`	→	Taylor's theorem with Lagrange remainder	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2445
`depends_on`	←	Causality implies dispersion relations	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2772
`depends_on`	←	Fundamental strip and holomorphy	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:3228
`proves`	←	ch:15-fourier-integral-transforms@proof-40	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2448