proposition 9.87 The distinguished limit, and which end carries the layer

open in the book · parts/02-mathematical-methods/07-odes-sturm-liouville.tex:3299 · p. 311

Rests on

depends_on equation 9.101 eq:slt-layer-equation ¶
depends_on proposition 9.18 The linear equation: integrating factor ¶
- 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 theorem 7.42 Fundamental theorem of calculus, I ¶
  - 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 7.40 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 ¶
- 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 ¶
      - proves proof ch:05-real-analysis@proof-20 ¶
    - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
  - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶ ↺
  - proves proof ch:05-real-analysis@proof-25 ¶
- proves proof ch:07-odes-sturm-liouville@proof-9 ¶
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 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-42 ¶

Supports

depends_on proposition 9.88 The model problem, and the error of the composite ¶

Neighborhood

Every logical edge within two steps of this node.

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partly declared
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Edges

type	direction	node	provenance	where
`depends_on`	→	eq:slt-layer-equation	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:3319
`depends_on`	→	The linear equation: integrating factor	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:3319
`depends_on`	→	Solution of a constant-coefficient system	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:3319
`depends_on`	←	The model problem, and the error of the composite	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:3371
`proves`	←	ch:07-odes-sturm-liouville@proof-42	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:3323