theorem 8.19 Fundamental theorem of algebra

open in the book · parts/02-mathematical-methods/06-complex-analysis.tex:480 · p. 270

Rests on

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 ¶
proves proof ch:06-complex-analysis@proof-13 ¶

Supports

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 definition 30.31 The engineering moduli ¶
      - depends_on phenomenon 30.70 Hertzian contact ¶
      - depends_on phenomenon 30.60 Coulomb's law of torsion ¶
      - depends_on proposition 30.33 Conversion among the moduli ¶
      - depends_on proposition 30.51 Moment and curvature ¶
      - depends_on remark A.676 The third cubic constant is measured, and it is not small ¶
      - depends_on theorem A.695 Boussinesq's point-force solution, quoted ¶
    - depends_on definition 30.79 Maxwell viscoelasticity ¶
      - depends_on proposition 30.80 A Maxwell material is a solid or a fluid depending on how long you watch ¶
    - depends_on proposition 30.74 The maximum shear stress, and its indifference to pressure ¶
      - depends_on phenomenon 30.73 Yield and brittle fracture ¶
      - depends_on remark A.705 The dangerous stress is below the surface ¶
      - depends_on remark 30.75 Two criteria, one uncited ¶
    - depends_on remark 30.75 Two criteria, one uncited ¶ ↺
  - depends_on proposition 30.9 Principal strains ¶
    - depends_on example 30.10 Simple shear and pure shear are the same strain ¶
  - depends_on theorem 5.80 Simultaneous diagonalization of commuting self-adjoint operators ¶
    - depends_on corollary 5.82 Spectral theorem for a normal operator ¶
  - depends_on theorem 5.84 Spectral theorem for a real symmetric operator ¶
    - depends_on corollary 5.91 The operator norm of a real array ¶
      - depends_on proposition 5.92 The eigenvalues do not control the norm ¶
    - depends_on lemma A.610 Block positivity ¶
      - depends_on proposition A.611 The constraint on $\Gamma$ ¶
    - depends_on lemma A.601 Several variables, complex symmetric matrix ¶
      - depends_on proposition A.612 The dictionary between $A$ and $\Gamma$ ¶
      - depends_on proposition A.602 The Wigner function of a Gaussian is a positive Gaussian ¶
    - depends_on lemma A.600 The real part of an inverse ¶
      - depends_on lemma A.601 Several variables, complex symmetric matrix ¶ ↺
      - depends_on proposition A.612 The dictionary between $A$ and $\Gamma$ ¶ ↺
    - depends_on proposition 5.90 Polar decomposition ¶
    - depends_on proposition 5.94 Principal axes of a real quadratic form ¶
      - depends_on proposition 5.95 The focal polar equation of a conic ¶
    - depends_on proposition 5.92 The eigenvalues do not control the norm ¶ ↺
    - depends_on theorem 5.86 Simultaneous diagonalization of a definite pencil ¶
      - depends_on corollary 5.89 Rayleigh–Ritz ¶
      - depends_on proposition 5.88 Rayleigh quotient of a definite pencil ¶
  - depends_on theorem 28.33 Normal modes ¶
    - depends_on example 28.39 The linear triatomic molecule ¶
    - depends_on proposition 28.40 Modes and dispersion of the loaded string ¶
    - depends_on remark 28.34 Part II owes the simultaneous diagonalization of two quadratic forms ¶
  - depends_on theorem 29.16 Principal axes ¶
    - depends_on definition 29.19 Inertia ellipsoid ¶
      - depends_on theorem 29.29 Poinsot's construction ¶
    - depends_on definition 29.35 The Lagrange top ¶
    - depends_on definition 29.18 Classification of tops ¶
      - depends_on proposition 29.28 Free rotation of a symmetric body ¶
    - depends_on phenomenon 29.31 The intermediate-axis instability ¶
      - depends_on remark 29.32 What the linearization is entitled to conclude ¶
    - depends_on proposition 29.33 Dissipation drives a free body to its largest moment ¶
    - depends_on proposition 29.17 Triangle inequalities ¶
      - depends_on proposition 29.22 Perpendicular-axis theorem ¶
    - depends_on theorem 29.25 Euler's equations ¶
      - depends_on phenomenon 29.46 A fast spinning top does not fall ¶
      - depends_on phenomenon 29.31 The intermediate-axis instability ¶ ↺
      - depends_on proposition 29.26 The two integrals of torque-free motion ¶
      - depends_on proposition 29.28 Free rotation of a symmetric body ¶ ↺
      - depends_on proposition 24.41 The free rigid body is a Lie–Poisson system ¶
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 corollary 15.46 The sign data, hypothesis (H3), and the count eleven ¶
    - depends_on theorem 15.43 The three constants of the classification are the three structure constants of $A$ ¶
      - depends_on corollary 15.46 The sign data, hypothesis (H3), and the count eleven ¶ ↺
      - depends_on corollary 15.44 The eight vertices are the eight degenerations of $A$ ¶
  - depends_on proposition 15.28 The quadratic family ¶
    - depends_on corollary 15.32 The expansion parameter is the cosmological constant ¶
      - depends_on example 15.64 The whole chapter at $D=4$ ¶
      - depends_on remark 15.63 The inclusion is strict, and the strictness is physical ¶
      - depends_on remark 15.33 Which branch nature is on ¶
    - depends_on remark 15.63 The inclusion is strict, and the strictness is physical ¶ ↺
    - depends_on theorem 15.29 The three constant-curvature kinematics, in every dimension ¶
      - depends_on corollary 15.32 The expansion parameter is the cosmological constant ¶ ↺
      - depends_on example 15.64 The whole chapter at $D=4$ ¶ ↺
  - 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 ¶

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Liouville	declared	parts/02-mathematical-methods/06-complex-analysis.tex:483
`depends_on`	←	Existence of an eigenvalue over $\C$	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:3243
`depends_on`	←	Artin decomposition	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:326
`depends_on`	←	The scalar constant-coefficient equation	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1024
`proves`	←	ch:06-complex-analysis@proof-13	declared	parts/02-mathematical-methods/06-complex-analysis.tex:486