theorem 8.17 Derivatives of all orders; Cauchy estimates

open in the book · parts/02-mathematical-methods/06-complex-analysis.tex:435 · p. 270

Rests on

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 definition 6.6 Continuous map ¶
      - depends_on definition 3.53 Preimage ¶
      - depends_on definition 6.2 Open set ¶
      - depends_on definition 6.1 Topological space ¶
    - depends_on definition 6.15 Path-connected space ¶
      - depends_on definition 6.6 Continuous map ¶ ↺
  - depends_on theorem 8.12 Cauchy ¶
    - depends_on equation 8.5 eq:cpx-cr ¶
    - depends_on theorem 7.131 Green ¶
      - depends_on definition 7.127 Simple regions ¶
      - depends_on remark 7.128 What the derivations below take as given ¶
      - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
      - proves proof ch:05-real-analysis@proof-77 ¶
    - proves proof ch:06-complex-analysis@proof-7 ¶
  - 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 ¶
    - depends_on equation 8.2 eq:cpx-exp-series ¶
    - depends_on proposition 7.47 Comparison; absolute convergence ¶
      - depends_on corollary A.46 Monotone convergence ¶
      - depends_on definition 7.45 Series ¶
      - depends_on theorem 7.8 Cauchy criterion ¶
      - proves proof ch:05-real-analysis@proof-28 ¶
    - proves proof ch:06-complex-analysis@proof-2 ¶
  - 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 ¶

Supports

depends_on theorem 8.18 Liouville ¶
- depends_on corollary A.723 Potential of one infinite row ¶
  - depends_on definition A.724 The staggered street ¶
    - depends_on proposition A.725 The street translates rigidly ¶
    - depends_on theorem A.729 Kármán's spacing ratio ¶
      - depends_on proposition A.730 The unstaggered street is always unstable ¶
  - depends_on proposition A.725 The street translates rigidly ¶ ↺
- depends_on lemma A.608 An entire function with a quadratic bound on its real part ¶
  - 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 $\Gamma$ ¶
- depends_on theorem 8.19 Fundamental theorem of algebra ¶
  - depends_on corollary 5.72 Existence of an eigenvalue over $\C$ ¶
    - depends_on theorem 5.98 Primary decomposition ¶
      - depends_on corollary 5.99 Semisimple and nilpotent parts ¶
    - depends_on theorem 5.79 Spectral theorem for a self-adjoint operator ¶
      - depends_on definition 30.20 Pressure and deviatoric stress ¶
      - depends_on proposition 30.9 Principal strains ¶
      - depends_on theorem 5.80 Simultaneous diagonalization of commuting self-adjoint operators ¶
      - depends_on theorem 5.84 Spectral theorem for a real symmetric operator ¶
      - depends_on theorem 28.33 Normal modes ¶
      - depends_on theorem 29.16 Principal axes ¶
  - depends_on lemma 15.12 Artin decomposition ¶
    - depends_on corollary 15.11 Semisimplicity criterion ¶
      - depends_on proposition 15.42 The $\Gamma$-graded commutative algebras ¶
      - depends_on proposition 15.28 The quadratic family ¶
      - depends_on theorem 15.13 Classification of $\mathfrak{g}_{A}$ by the algebra ¶
    - depends_on theorem 15.13 Classification of $\mathfrak{g}_{A}$ by the algebra ¶ ↺
    - depends_on theorem 15.68 Positive Gram forms are positive sums of real characters ¶
      - depends_on corollary 15.69 No positive pairing survives the radical ¶
  - depends_on proposition 9.28 The scalar constant-coefficient equation ¶
    - depends_on proposition 9.88 The model problem, and the error of the composite ¶
- depends_on theorem 12.54 The spectrum is compact and non-empty ¶
  - depends_on theorem 12.55 The spectrum of a self-adjoint operator is real ¶
    - depends_on example 12.56 Multiplication by the coordinate: spectrum without eigenvectors ¶
    - depends_on theorem A.238 Spectral theorem, both forms ¶
      - depends_on proposition A.280 Direct-integral form of the spectral theorem ¶
      - depends_on proposition A.261 Spectral theorem for a unitary operator ¶
      - depends_on theorem A.279 Gelfand–Maurin ¶
      - depends_on theorem A.253 Stone ¶
    - depends_on theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶
      - depends_on definition 12.60 Functional calculus ¶
      - depends_on theorem 12.107 Nuclear spectral theorem ¶
      - depends_on theorem 12.91 Schur's lemma, commutant form ¶
      - depends_on theorem 12.66 Stone ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
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not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	ML estimate	declared	parts/02-mathematical-methods/06-complex-analysis.tex:445
`depends_on`	→	Cauchy integral formula	declared	parts/02-mathematical-methods/06-complex-analysis.tex:445
`depends_on`	←	Liouville	declared	parts/02-mathematical-methods/06-complex-analysis.tex:468
`proves`	←	ch:06-complex-analysis@proof-11	declared	parts/02-mathematical-methods/06-complex-analysis.tex:448