proposition A.609 The exponent is a quadratic polynomial

open in the book · appendices/A-long-proofs.tex:29158 · p. 3083

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depends_on lemma A.607 A zero-free entire function is an exponential ¶
- depends_on lemma A.606 $F$ is entire, and is the overlap in disguise ¶
  - depends_on definition A.605 The analytic continuation of the overlaps ¶
    - depends_on definition 25.52 Wigner function ¶
      - depends_on definition 17.29 Fourier transform; the treatise convention ¶
      - depends_on definition 25.44 Weyl and normal ordering ¶
    - depends_on equation A.918 eq:app-hudson-gaussian-coherent ¶
  - depends_on proposition 12.4 Cauchy–Schwarz and continuity of the inner product ¶
    - depends_on definition 12.2 Hilbert space ¶
      - depends_on definition 5.18 Inner product ¶
      - depends_on definition 6.27 Convergence; Cauchy sequence; completeness ¶
      - depends_on equation 5.44 eq:lin-norm-assoc ¶
    - depends_on proposition 5.20 Cauchy–Schwarz inequality ¶
      - depends_on definition 5.14 Linear independence ¶
      - depends_on definition 5.18 Inner product ¶ ↺
      - proves proof ch:03-linear-algebra-representations@proof-4 ¶
    - proves proof ch:10-hilbert-spaces@proof-1 ¶
  - depends_on theorem 8.6 Cauchy–Riemann equations ¶
    - 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 equation 8.4 eq:cpx-derivative ¶
    - proves proof ch:06-complex-analysis@proof-3 ¶
  - proves proof app:A-long-proofs@proof-364 ¶
- depends_on theorem 7.109 Leibniz integral rule ¶
  - 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 axiom 7.1 Completeness of $\R$ ¶ ↺
  - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
    - 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-10 ¶
  - depends_on theorem 7.35 Mean value theorem ¶
    - depends_on proposition 7.29 Linearity ¶
      - depends_on definition 7.11 Sum of functions ¶
      - depends_on equation 7.14 eq:ana-derivh ¶
      - depends_on proposition 7.6 Algebra of limits ¶
      - proves proof ch:05-real-analysis@proof-13 ¶
    - depends_on theorem 7.34 Rolle ¶
      - depends_on lemma 7.33 Fermat: interior extremum ¶
      - depends_on theorem 7.24 Extreme value theorem ¶
      - proves proof ch:05-real-analysis@proof-18 ¶
    - proves proof ch:05-real-analysis@proof-19 ¶
  - proves proof ch:05-real-analysis@proof-67 ¶
- depends_on theorem 8.6 Cauchy–Riemann equations ¶ ↺
- proves proof app:A-long-proofs@proof-365 ¶
depends_on lemma A.608 An entire function with a quadratic bound on its real part ¶
- depends_on equation 8.13 eq:cpx-derivative-formula ¶
- depends_on theorem 8.18 Liouville ¶
  - depends_on proposition 8.10 Fundamental theorem for contours ¶
    - depends_on equation 8.6 eq:cpx-contour-integral ¶
    - 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:06-complex-analysis@proof-5 ¶
  - depends_on theorem 8.17 Derivatives of all orders; Cauchy estimates ¶
    - 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 theorem 8.16 Cauchy integral formula ¶
      - depends_on corollary 8.15 Deformation of contours ¶
      - depends_on lemma 8.11 The fundamental $2\pi\ii$ ¶
      - depends_on lemma 8.9 ML estimate ¶ ↺
      - proves proof ch:06-complex-analysis@proof-10 ¶
    - proves proof ch:06-complex-analysis@proof-11 ¶
  - proves proof ch:06-complex-analysis@proof-12 ¶
- depends_on theorem 8.20 Taylor expansion ¶
  - depends_on equation 8.12 eq:cpx-cif ¶
  - depends_on equation 8.13 eq:cpx-derivative-formula ¶ ↺
  - depends_on lemma 8.9 ML estimate ¶ ↺
  - depends_on proposition 7.46 Geometric series ¶
    - depends_on corollary A.46 Monotone convergence ¶ ↺
    - depends_on definition 7.45 Series ¶
      - depends_on definition 7.2 Absolute value ¶
      - depends_on definition 7.4 Convergence ¶ ↺
    - proves proof ch:05-real-analysis@proof-27 ¶
  - proves proof ch:06-complex-analysis@proof-14 ¶
- proves proof app:A-long-proofs@proof-366 ¶
depends_on proposition A.604 Non-negativity forbids a vanishing overlap ¶
- depends_on equation 25.54 eq:pq-wigner-marginal-x ¶
- depends_on lemma A.603 Overlaps and the resolution of unity ¶
  - depends_on equation A.918 eq:app-hudson-gaussian-coherent ¶ ↺
  - depends_on equation 25.56 eq:pq-wigner-overlap ¶
  - depends_on theorem 25.53 Properties of the Wigner function ¶
    - depends_on equation 25.53 eq:pq-wigner ¶
    - depends_on proposition 12.4 Cauchy–Schwarz and continuity of the inner product ¶ ↺
    - 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:08-poisson-quantum-bridge@proof-20 ¶
  - proves proof app:A-long-proofs@proof-362 ¶
- depends_on proposition A.602 The Wigner function of a Gaussian is a positive Gaussian ¶
  - depends_on equation A.910 eq:app-hudson-gaussian-form ¶
  - depends_on equation 25.53 eq:pq-wigner ¶ ↺
  - depends_on lemma A.601 Several variables, complex symmetric matrix ¶
    - depends_on lemma A.599 One variable, complex coefficient ¶
      - depends_on equation 17.42 eq:ft-gaussian ¶ ↺
      - depends_on theorem 7.109 Leibniz integral rule ¶ ↺
      - depends_on theorem 8.6 Cauchy–Riemann equations ¶ ↺
      - proves proof app:A-long-proofs@proof-358 ¶
    - depends_on lemma A.600 The real part of an inverse ¶
      - depends_on theorem 5.40 Rank–nullity ¶
      - depends_on theorem 5.84 Spectral theorem for a real symmetric operator ¶
      - proves proof app:A-long-proofs@proof-359 ¶
    - depends_on theorem 5.84 Spectral theorem for a real symmetric operator ¶ ↺
    - proves proof app:A-long-proofs@proof-360 ¶
  - proves proof app:A-long-proofs@proof-361 ¶
- proves proof app:A-long-proofs@proof-363 ¶
proves proof app:A-long-proofs@proof-367 ¶

Supports

depends_on proposition A.611 The constraint on $\Gamma$ ¶
- depends_on proposition A.612 The dictionary between $A$ and $\Gamma$ ¶

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`	→	A zero-free entire function is an exponential	declared	appendices/A-long-proofs.tex:29168
`depends_on`	→	An entire function with a quadratic bound on its real part	declared	appendices/A-long-proofs.tex:29168
`depends_on`	→	Non-negativity forbids a vanishing overlap	declared	appendices/A-long-proofs.tex:29168
`depends_on`	←	The constraint on $\Gamma$	declared	appendices/A-long-proofs.tex:29285
`proves`	←	app:A-long-proofs@proof-367	declared	appendices/A-long-proofs.tex:29172