theorem 17.93 Fundamental strip and holomorphy

open in the book · parts/02-mathematical-methods/15-fourier-integral-transforms.tex:3217 · p. 708

Rests on

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 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 ¶
proves proof ch:15-fourier-integral-transforms@proof-56 ¶

Supports

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`	→	Mellin transform	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:3228
`depends_on`	→	Holomorphy	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:3228
`depends_on`	←	Mellin inversion	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:3258
`proves`	←	ch:15-fourier-integral-transforms@proof-56	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:3231