proposition 9.142 The differential equation of the sine amplitude

Rests on

• depends_on proposition 9.141 First properties ¶
  • depends_on definition 9.140 Amplitude and the Jacobi elliptic functions ¶
    • depends_on definition 9.137 Elliptic integrals of the three kinds ¶
      • depends_on definition 7.125 Multiple integral ¶
        • depends_on axiom 7.1 Completeness of $\R$ ¶
        • depends_on definition 7.39 Darboux sums and the definite integral ¶
      • 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 proposition 7.32 Derivative of the inverse function ¶
      • depends_on definition 7.14 Inverse function ¶
        • depends_on definition 7.13 Composite function ¶
        • depends_on definition 3.47 Bijective map ¶
      • 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 ¶
      • depends_on theorem 7.23 Intermediate value theorem ¶
        • depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
        • depends_on definition 7.20 Continuity at a point ¶
        • proves proof ch:05-real-analysis@proof-8 ¶
      • proves proof ch:05-real-analysis@proof-16 ¶
  • depends_on lemma 7.71 Derivatives; the Pythagorean identity ¶
    • depends_on equation 7.53 eq:ana-cos-sin-series ¶
    • depends_on equation 5.3 eq:lin-binomial ¶
    • depends_on proposition 7.49 Cauchy product ¶
      • depends_on corollary A.46 Monotone convergence ¶
        • depends_on theorem A.40 Least-upper-bound property ¶
        • proves proof app:A-long-proofs@proof-33 ¶
      • depends_on definition 7.45 Series ¶
        • depends_on definition 7.2 Absolute value ¶
        • depends_on definition 7.4 Convergence ¶ ↺
      • depends_on proposition 7.6 Algebra of limits ¶ ↺
      • proves proof ch:05-real-analysis@proof-30 ¶
    • depends_on theorem 7.51 Termwise differentiation ¶
      • depends_on proposition 7.47 Comparison; absolute convergence ¶
        • depends_on corollary A.46 Monotone convergence ¶ ↺
        • depends_on definition 7.45 Series ¶ ↺
        • depends_on theorem 7.8 Cauchy criterion ¶
        • proves proof ch:05-real-analysis@proof-28 ¶
      • 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:05-real-analysis@proof-46 ¶
  • depends_on proposition 7.32 Derivative of the inverse function ¶ ↺
  • proves proof ch:07-odes-sturm-liouville@proof-75 ¶
• 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-76 ¶

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