proposition 9.28 The scalar constant-coefficient equation

open in the book · parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1003 · p. 284

Rests on

depends_on proposition 7.30 Leibniz rule ¶
- depends_on definition 7.12 Product of functions ¶
  - depends_on definition 7.9 Real function ¶
    - depends_on definition 3.43 Map ¶
      - depends_on definition 3.24 Quantifiers ¶
      - depends_on definition 3.28 Set ¶
- depends_on equation 7.14 eq:ana-derivh ¶
- depends_on proposition 7.27 Differentiable implies continuous ¶
  - depends_on definition 7.20 Continuity 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 equation 7.7 eq:ana-limit-left ¶
    - depends_on equation 7.5 eq:ana-limit-right ¶
  - depends_on definition 7.26 Derivative of a function at a point ¶
    - depends_on definition 7.16 Limit ¶ ↺
    - depends_on definition 7.9 Real function ¶ ↺
  - depends_on proposition 7.6 Algebra of limits ¶
    - depends_on definition 7.4 Convergence ¶
      - depends_on definition 7.2 Absolute value ¶ ↺
    - depends_on proposition 7.3 Triangle inequality ¶
      - depends_on definition 7.2 Absolute value ¶ ↺
      - proves proof ch:05-real-analysis@proof-1 ¶
    - proves proof ch:05-real-analysis@proof-3 ¶
  - proves proof ch:05-real-analysis@proof-11 ¶
- depends_on proposition 7.6 Algebra of limits ¶ ↺
- proves proof ch:05-real-analysis@proof-14 ¶
depends_on theorem 8.19 Fundamental theorem of algebra ¶
- 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 theorem 7.42 Fundamental theorem of calculus, I ¶
      - 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 lemma 8.11 The fundamental $2\pi\ii$ ¶
      - 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 ¶
depends_on theorem 9.13 Dimension of the solution space ¶
- 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 definition 9.2 Linear equation; homogeneity ¶ ↺
- proves proof ch:07-odes-sturm-liouville@proof-6 ¶
proves proof ch:07-odes-sturm-liouville@proof-16 ¶

Supports

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

Neighborhood

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partly declared
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declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Leibniz rule	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1024
`depends_on`	→	Fundamental theorem of algebra	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1024
`depends_on`	→	Dimension of the solution space	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1024
`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-16	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1027