lemma A.608 An entire function with a quadratic bound on its real part

open in the book · appendices/A-long-proofs.tex:29099 · p. 3083

Rests on

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 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: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 definition 6.17 Simply connected space ¶
      - depends_on theorem 8.12 Cauchy ¶
      - proves proof ch:06-complex-analysis@proof-9 ¶
    - depends_on lemma 8.11 The fundamental $2\pi\ii$ ¶
      - depends_on equation 8.6 eq:cpx-contour-integral ¶ ↺
      - depends_on proposition 8.4 Euler's formula ¶
      - proves proof ch:06-complex-analysis@proof-6 ¶
    - 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 theorem A.40 Least-upper-bound property ¶
      - depends_on definition A.36 Cut ¶
      - depends_on definition A.38 Order on $\R$ ¶
      - proves proof app:A-long-proofs@proof-29 ¶
    - proves proof app:A-long-proofs@proof-33 ¶
  - depends_on definition 7.45 Series ¶
    - depends_on definition 7.2 Absolute value ¶
    - depends_on definition 7.4 Convergence ¶
      - depends_on definition 7.2 Absolute value ¶ ↺
  - proves proof ch:05-real-analysis@proof-27 ¶
- proves proof ch:06-complex-analysis@proof-14 ¶
proves proof app:A-long-proofs@proof-366 ¶

Supports

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`	→	eq:cpx-derivative-formula	declared	appendices/A-long-proofs.tex:29104
`depends_on`	→	Liouville	declared	appendices/A-long-proofs.tex:29104
`depends_on`	→	Taylor expansion	declared	appendices/A-long-proofs.tex:29104
`depends_on`	←	The exponent is a quadratic polynomial	declared	appendices/A-long-proofs.tex:29168
`proves`	←	app:A-long-proofs@proof-366	declared	appendices/A-long-proofs.tex:29107