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

Rests on

• depends_on proposition 9.28 The scalar constant-coefficient equation ¶
  • 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 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 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 proposition 7.3 Triangle inequality ¶
        • 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 ¶
        • 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 theorem 8.16 Cauchy integral formula ¶
        • 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 ¶
        • 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 ¶
• depends_on proposition 9.87 The distinguished limit, and which end carries the layer ¶
  • 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 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 theorem 7.24 Extreme value theorem ¶
        • depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
        • 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 ¶ ↺
    • 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 proposition 5.125 prop:lin-matrix-algebra ¶
      • 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 proposition 7.48 Ratio test ¶
        • depends_on proposition 7.6 Algebra of limits ¶ ↺
        • depends_on theorem 7.50 Power series; radius of convergence ¶
        • 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 ¶
• depends_on theorem 7.38 Taylor's theorem with Lagrange remainder ¶
  • depends_on proposition 7.30 Leibniz rule ¶ ↺
  • depends_on theorem 7.34 Rolle ¶
    • depends_on lemma 7.33 Fermat: interior extremum ¶
      • depends_on definition 7.26 Derivative of a function at a point ¶ ↺
      • depends_on proposition 7.18 Two-sided limit from one-sided limits ¶
        • depends_on definition 7.16 Limit ¶ ↺
        • depends_on equation 7.7 eq:ana-limit-left ¶ ↺
        • depends_on equation 7.5 eq:ana-limit-right ¶ ↺
        • proves proof ch:05-real-analysis@proof-6 ¶
      • proves proof ch:05-real-analysis@proof-17 ¶
    • depends_on theorem 7.24 Extreme value theorem ¶ ↺
    • proves proof ch:05-real-analysis@proof-18 ¶
  • proves proof ch:05-real-analysis@proof-22 ¶
• proves proof ch:07-odes-sturm-liouville@proof-43 ¶

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