proposition 17.41 The identities physics uses

open in the book · parts/02-mathematical-methods/15-fourier-integral-transforms.tex:1592 · p. 689

Rests on

depends_on definition 17.40 Transform and derivative of a distribution ¶
- depends_on definition 17.38 Schwartz space and tempered distributions ¶
- depends_on proposition 17.39 The transform preserves $\mathcal{S}$ ¶
  - depends_on definition 17.38 Schwartz space and tempered distributions ¶ ↺
  - depends_on proposition 17.30 Elementary properties ¶
    - depends_on definition 17.29 Fourier transform; the treatise convention ¶
    - depends_on equation 7.28 eq:ana-parts ¶
    - depends_on lemma 17.7 Riemann–Lebesgue ¶
      - depends_on definition 7.39 Darboux sums and the definite integral ¶
      - proves proof ch:15-fourier-integral-transforms@proof-5 ¶
    - proves proof ch:15-fourier-integral-transforms@proof-19 ¶
  - 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-25 ¶
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 theorem 7.35 Mean value theorem ¶
      - depends_on proposition 7.29 Linearity ¶
      - depends_on theorem 7.34 Rolle ¶
      - proves proof ch:05-real-analysis@proof-19 ¶
    - proves proof ch:05-real-analysis@proof-20 ¶
  - depends_on definition 7.26 Derivative of a function at a point ¶
    - depends_on definition 7.16 Limit ¶
      - depends_on definition 7.2 Absolute value ¶
      - depends_on definition 7.9 Real function ¶
    - depends_on definition 7.9 Real function ¶ ↺
- 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 definition 7.39 Darboux sums and the definite integral ¶ ↺
    - 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 ¶
    - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
      - depends_on proposition 7.22 Sequential characterization ¶ ↺
      - depends_on theorem 7.7 Bolzano–Weierstrass ¶ ↺
      - proves proof ch:05-real-analysis@proof-10 ¶
    - proves proof ch:05-real-analysis@proof-23 ¶
  - depends_on theorem 7.24 Extreme value theorem ¶ ↺
  - proves proof ch:05-real-analysis@proof-24 ¶
- proves proof ch:05-real-analysis@proof-25 ¶
depends_on theorem 17.32 Fourier inversion ¶ ↺
proves proof ch:15-fourier-integral-transforms@proof-26 ¶

Supports

depends_on proposition 17.90 Hankel transform as the axially symmetric Fourier transform ¶
depends_on proposition 17.52 Linear translation-invariant systems ¶
- depends_on definition 28.25 Complex susceptibility ¶
- depends_on proposition 17.61 Mean-square response of a damped resonator ¶

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`	→	Transform and derivative of a distribution	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:1611
`depends_on`	→	Fundamental theorem of calculus, II	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:1611
`depends_on`	→	Fourier inversion	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:1611
`depends_on`	←	Hankel transform as the axially symmetric Fourier transform	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:3157
`depends_on`	←	Linear translation-invariant systems	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:1992
`proves`	←	ch:15-fourier-integral-transforms@proof-26	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:1614