proposition 8.10 Fundamental theorem for contours

open in the book · parts/02-mathematical-methods/06-complex-analysis.tex:209 · p. 267

Rests on

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 theorem 7.35 Mean value theorem ¶
      - depends_on proposition 7.29 Linearity ¶
      - depends_on theorem 7.34 Rolle ¶
      - proves proof ch:05-real-analysis@proof-19 ¶
    - proves proof ch:05-real-analysis@proof-20 ¶
  - depends_on definition 7.26 Derivative of a function 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 definition 7.9 Real function ¶ ↺
- 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 definition 7.39 Darboux sums and the definite integral ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶
    - 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 ¶
    - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
      - depends_on proposition 7.22 Sequential characterization ¶ ↺
      - depends_on theorem 7.7 Bolzano–Weierstrass ¶ ↺
      - proves proof ch:05-real-analysis@proof-10 ¶
    - proves proof ch:05-real-analysis@proof-23 ¶
  - 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 ¶

Supports

depends_on theorem 8.13 Goursat ¶
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

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Edges

type	direction	node	provenance	where
`depends_on`	→	eq:cpx-contour-integral	declared	parts/02-mathematical-methods/06-complex-analysis.tex:215
`depends_on`	→	Fundamental theorem of calculus, II	declared	parts/02-mathematical-methods/06-complex-analysis.tex:215
`depends_on`	←	Goursat	declared	parts/02-mathematical-methods/06-complex-analysis.tex:301
`depends_on`	←	Liouville	declared	parts/02-mathematical-methods/06-complex-analysis.tex:468
`proves`	←	ch:06-complex-analysis@proof-5	declared	parts/02-mathematical-methods/06-complex-analysis.tex:218