theorem 9.26 Structure of the solutions

open in the book · parts/02-mathematical-methods/07-odes-sturm-liouville.tex:876 · p. 282

Rests on

depends_on theorem 8.24 Residue theorem ¶
- 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 definition 8.22 Isolated singularities; residue ¶
  - 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 ¶
      - proves proof ch:06-complex-analysis@proof-2 ¶
    - proves proof ch:06-complex-analysis@proof-6 ¶
  - depends_on theorem 8.21 Laurent expansion ¶
    - depends_on corollary 8.15 Deformation of contours ¶ ↺
    - depends_on lemma 8.11 The fundamental $2\pi\ii$ ¶ ↺
    - depends_on theorem 8.16 Cauchy integral formula ¶
      - depends_on corollary 8.15 Deformation of contours ¶ ↺
      - depends_on lemma 8.11 The fundamental $2\pi\ii$ ¶ ↺
      - depends_on lemma 8.9 ML estimate ¶
      - proves proof ch:06-complex-analysis@proof-10 ¶
    - 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 ¶
      - proves proof ch:06-complex-analysis@proof-14 ¶
    - proves proof ch:06-complex-analysis@proof-15 ¶
- depends_on lemma 8.11 The fundamental $2\pi\ii$ ¶ ↺
- depends_on theorem 8.21 Laurent expansion ¶ ↺
- proves proof ch:06-complex-analysis@proof-17 ¶
depends_on theorem 5.71 The eigenvalues are the roots of the characteristic polynomial ¶
- depends_on definition 5.70 Characteristic polynomial ¶
  - depends_on definition 5.56 Endomorphism ¶
    - depends_on definition 5.37 Linear transformation ¶
      - depends_on definition 4.33 Vector space ¶
  - depends_on equation 5.19 eq:lin-leibniz-det ¶
  - depends_on equation 5.74 eq:lin-matrix-rep ¶
  - proves proof ch:03-linear-algebra-representations@proof-25 ¶
- depends_on definition 5.69 Eigenvector, eigenvalue, eigenspace ¶
  - depends_on definition 5.56 Endomorphism ¶ ↺
  - depends_on definition 5.39 Kernel, image, nullity, rank ¶
    - depends_on definition 5.37 Linear transformation ¶ ↺
    - depends_on definition 5.7 Vector subspace ¶
      - depends_on definition 5.5 Linear combination ¶
    - proves proof ch:03-linear-algebra-representations@proof-12 ¶
- depends_on proposition 5.49 Injective, surjective, invertible ¶
  - depends_on definition 5.39 Kernel, image, nullity, rank ¶ ↺
  - depends_on proposition 5.48 Existence, uniqueness, and linearity of the inverse ¶
    - depends_on definition 5.47 Inverse of a linear transformation ¶
      - depends_on definition 5.37 Linear transformation ¶ ↺
    - depends_on definition 5.37 Linear transformation ¶ ↺
    - proves proof ch:03-linear-algebra-representations@proof-17 ¶
  - depends_on theorem 5.40 Rank–nullity ¶
    - 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 definition 5.39 Kernel, image, nullity, rank ¶ ↺
    - depends_on lemma 5.38 Exchange and completion ¶
      - 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 ¶ ↺
      - proves proof ch:03-linear-algebra-representations@proof-11 ¶
    - proves proof ch:03-linear-algebra-representations@proof-13 ¶
  - proves proof ch:03-linear-algebra-representations@proof-18 ¶
- proves proof ch:03-linear-algebra-representations@proof-26 ¶
depends_on theorem 9.23 Solution of a constant-coefficient system ¶
- depends_on corollary 9.9 Linear equations: existence on the whole interval ¶
  - depends_on definition 9.2 Linear equation; homogeneity ¶
    - depends_on definition 9.1 Ordinary differential equation ¶
  - depends_on remark 9.3 Normal form and first-order systems ¶
  - depends_on theorem 9.8 Picard–Lindelöf ¶
    - depends_on definition 9.4 Lipschitz condition ¶
    - depends_on equation 9.3 eq:slt-ode-system ¶
    - depends_on lemma 9.6 Weierstrass $M$-test; uniform limits are continuous ¶
      - depends_on definition 7.20 Continuity at a point ¶
      - depends_on proposition 7.47 Comparison; absolute convergence ¶ ↺
      - proves proof ch:07-odes-sturm-liouville@proof-1 ¶
    - proves proof ch:07-odes-sturm-liouville@proof-2 ¶
  - proves proof ch:07-odes-sturm-liouville@proof-3 ¶
- depends_on proposition 9.22 The exponential and its derivative ¶
  - depends_on definition 9.21 Matrix exponential ¶
    - depends_on definition 5.19 Norm ¶
      - depends_on definition 4.33 Vector space ¶ ↺
    - depends_on proposition 5.125 prop:lin-matrix-algebra ¶
      - depends_on definition 5.121 Algebra ¶
      - depends_on definition 5.37 Linear transformation ¶ ↺
      - depends_on equation 5.150 eq:lin-ctesesc2 ¶
      - depends_on equation 5.74 eq:lin-matrix-rep ¶ ↺
      - depends_on proposition 5.45 prop:lin-matrix-unique ¶
      - proves proof ch:03-linear-algebra-representations@proof-55 ¶
  - depends_on lemma 9.6 Weierstrass $M$-test; uniform limits are continuous ¶ ↺
  - depends_on theorem 7.51 Termwise differentiation ¶
    - depends_on proposition 7.47 Comparison; absolute convergence ¶ ↺
    - depends_on proposition 7.27 Differentiable implies continuous ¶
      - depends_on definition 7.20 Continuity at a point ¶ ↺
      - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on proposition 7.6 Algebra of limits ¶
      - proves proof ch:05-real-analysis@proof-11 ¶
    - depends_on proposition 7.48 Ratio test ¶
      - depends_on proposition 7.47 Comparison; absolute convergence ¶ ↺
      - depends_on proposition 7.46 Geometric series ¶ ↺
      - depends_on theorem 7.8 Cauchy criterion ¶
      - proves proof ch:05-real-analysis@proof-29 ¶
    - depends_on proposition 7.6 Algebra of limits ¶ ↺
    - depends_on theorem 7.50 Power series; radius of convergence ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶
      - depends_on proposition 7.47 Comparison; absolute convergence ¶ ↺
      - depends_on proposition 7.46 Geometric series ¶ ↺
      - proves proof ch:05-real-analysis@proof-31 ¶
    - proves proof ch:05-real-analysis@proof-32 ¶
  - proves proof ch:07-odes-sturm-liouville@proof-11 ¶
- proves proof ch:07-odes-sturm-liouville@proof-12 ¶
proves proof ch:07-odes-sturm-liouville@proof-14 ¶

Supports

depends_on corollary 9.27 Linear stability ¶
- depends_on corollary 9.41 Hill's equation ¶
  - depends_on proposition 9.43 The unmodulated line, and the feet of the tongues ¶
- depends_on proposition 9.31 Lyapunov's first method: asymptotic stability ¶
- depends_on proposition 9.29 The planar classification ¶
- depends_on theorem 9.39 Floquet ¶
  - depends_on corollary 9.41 Hill's equation ¶ ↺

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Residue theorem	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:891
`depends_on`	→	The eigenvalues are the roots of the characteristic polynomial	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:891
`depends_on`	→	Solution of a constant-coefficient system	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:891
`depends_on`	←	Linear stability	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:974
`proves`	←	ch:07-odes-sturm-liouville@proof-14	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:894