lemma A.606 $F$ is entire, and is the overlap in disguise

open in the book · appendices/A-long-proofs.tex:28994 · p. 3082

Rests on

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 definition 25.42 Ordering rule ¶
      - depends_on corollary 25.41 Quantization is not a functor ¶
      - depends_on postulate 25.27 Dirac's correspondence rule ¶
- 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 4.33 Vector space ¶
      - depends_on definition 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.32 Field ¶
      - depends_on definition 4.31 Module ¶
  - depends_on definition 6.27 Convergence; Cauchy sequence; completeness ¶
    - depends_on definition 6.24 Metric ¶
      - depends_on definition 3.43 Map ¶
      - depends_on definition 3.38 Ordered pair and Cartesian product ¶
  - 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 5.5 Linear combination ¶
      - depends_on definition 4.33 Vector space ¶ ↺
  - 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 7.16 Limit ¶
      - depends_on definition 7.2 Absolute value ¶
      - depends_on definition 7.9 Real function ¶
    - depends_on equation 7.7 eq:ana-limit-left ¶
    - depends_on equation 7.5 eq:ana-limit-right ¶
  - depends_on definition 7.97 Partial derivative; gradient ¶
    - 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 definition 5.15 Basis ¶
      - depends_on definition 5.12 Subspace generated by a set of vectors ¶
      - depends_on definition 5.14 Linear independence ¶ ↺
- depends_on equation 8.4 eq:cpx-derivative ¶
- proves proof ch:06-complex-analysis@proof-3 ¶
proves proof app:A-long-proofs@proof-364 ¶

Supports

depends_on lemma A.607 A zero-free entire function is an exponential ¶
- depends_on proposition A.609 The exponent is a quadratic polynomial ¶
  - 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.

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`	→	The analytic continuation of the overlaps	declared	appendices/A-long-proofs.tex:29014
`depends_on`	→	Cauchy–Schwarz and continuity of the inner product	declared	appendices/A-long-proofs.tex:29014
`depends_on`	→	Cauchy–Riemann equations	declared	appendices/A-long-proofs.tex:29014
`depends_on`	←	A zero-free entire function is an exponential	declared	appendices/A-long-proofs.tex:29067
`proves`	←	app:A-long-proofs@proof-364	declared	appendices/A-long-proofs.tex:29018