proposition 9.144 The pendulum equation at finite amplitude

Rests on

• depends_on corollary 7.44 Substitution and integration by parts ¶
  • depends_on proposition 7.31 Chain rule ¶
    • depends_on definition 7.13 Composite function ¶
      • depends_on definition 7.9 Real function ¶
        • depends_on definition 3.43 Map ¶
    • 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-15 ¶
  • depends_on proposition 7.30 Leibniz rule ¶
    • depends_on definition 7.12 Product of functions ¶
      • depends_on definition 7.9 Real function ¶ ↺
    • depends_on equation 7.14 eq:ana-derivh ¶
    • depends_on proposition 7.27 Differentiable implies continuous ¶ ↺
    • depends_on proposition 7.6 Algebra of limits ¶ ↺
    • proves proof ch:05-real-analysis@proof-14 ¶
  • 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 ¶
      • 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 ¶
    • proves proof ch:05-real-analysis@proof-25 ¶
  • proves proof ch:05-real-analysis@proof-26 ¶
• 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 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 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 proposition 9.139 Series for the complete integral of the first kind ¶
  • depends_on definition 9.137 Elliptic integrals of the three kinds ¶ ↺
  • 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 ¶
      • 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 theorem 7.8 Cauchy criterion ¶
        • depends_on corollary A.47 Cauchy completeness ¶
        • depends_on definition 7.4 Convergence ¶ ↺
        • proves proof ch:05-real-analysis@proof-5 ¶
      • proves proof ch:05-real-analysis@proof-28 ¶
    • proves proof ch:07-odes-sturm-liouville@proof-1 ¶
  • depends_on lemma 9.136 Wallis integrals ¶
    • depends_on corollary 7.44 Substitution and integration by parts ¶ ↺
    • 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 definition 7.45 Series ¶ ↺
        • 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 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:05-real-analysis@proof-46 ¶
    • proves proof ch:07-odes-sturm-liouville@proof-73 ¶
  • proves proof ch:07-odes-sturm-liouville@proof-74 ¶
• proves proof ch:07-odes-sturm-liouville@proof-78 ¶

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