theorem 17.78 Causality implies dispersion relations

open in the book · parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2756 · p. 703

Rests on

depends_on definition 17.29 Fourier transform; the treatise convention ¶
depends_on equation 8.16 eq:cpx-simple-pole ¶
depends_on lemma 8.9 ML estimate ¶
- depends_on equation 8.6 eq:cpx-contour-integral ¶
- proves proof ch:06-complex-analysis@proof-4 ¶
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 definition 7.12 Product of functions ¶
      - depends_on definition 7.9 Real function ¶
    - 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 7.26 Derivative of a function at a point ¶
      - depends_on proposition 7.6 Algebra of limits ¶
      - 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 ¶
      - 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 theorem 7.7 Bolzano–Weierstrass ¶
      - 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 ¶
depends_on proposition 17.42 Sokhotski–Plemelj ¶
- depends_on definition 17.38 Schwartz space and tempered distributions ¶
- 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-27 ¶
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 7.20 Continuity at a point ¶ ↺
      - depends_on definition 7.97 Partial derivative; gradient ¶
    - depends_on definition 6.9 Compact set ¶
      - depends_on definition 6.5 Open cover ¶
  - depends_on remark 7.128 What the derivations below take as given ¶
    - depends_on definition 7.125 Multiple integral ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
      - depends_on definition 7.39 Darboux sums and the definite integral ¶
    - depends_on theorem 7.40 Continuous functions are integrable ¶
      - depends_on definition 7.39 Darboux sums and the definite integral ¶ ↺
      - depends_on theorem 7.24 Extreme value theorem ¶ ↺
      - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
      - proves proof ch:05-real-analysis@proof-23 ¶
    - 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 corollary 7.36 cor:ana-mvt-consequences ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
    - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
      - depends_on theorem 7.40 Continuous functions are integrable ¶ ↺
      - depends_on theorem 7.24 Extreme value theorem ¶ ↺
      - proves proof ch:05-real-analysis@proof-24 ¶
    - proves proof ch:05-real-analysis@proof-25 ¶
  - proves proof ch:05-real-analysis@proof-77 ¶
- proves proof ch:06-complex-analysis@proof-7 ¶
proves proof ch:15-fourier-integral-transforms@proof-49 ¶

Supports

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 ¶

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`	→	Fourier transform; the treatise convention	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2772
`depends_on`	→	eq:cpx-simple-pole	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2772
`depends_on`	→	ML estimate	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2772
`depends_on`	→	Holomorphy	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2772
`depends_on`	→	Sokhotski–Plemelj	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2772
`depends_on`	→	Cauchy	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2772
`depends_on`	←	Fibre anelasticity, and why it biases one method and not the other	declared	parts/03-classical-mechanics/16-exp-cavendish.tex:593
`depends_on`	←	Causality is visible in the susceptibility	declared	parts/03-classical-mechanics/11-oscillations-waves.tex:785
`depends_on`	←	Neither velocity is a speed limit	declared	parts/03-classical-mechanics/11-oscillations-waves.tex:2251
`proves`	←	ch:15-fourier-integral-transforms@proof-49	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2776