proposition 9.139 Series for the complete integral of the first kind

open in the book · parts/02-mathematical-methods/07-odes-sturm-liouville.tex:5269 · p. 333

Rests on

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 axiom 7.1 Completeness of $\R$ ¶ ↺
- 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 axiom 7.1 Completeness of $\R$ ¶ ↺
    - depends_on proposition 7.22 Sequential characterization ¶
      - depends_on definition 7.20 Continuity at a point ¶
      - depends_on definition 7.4 Convergence ¶
      - proves proof ch:05-real-analysis@proof-7 ¶
    - depends_on theorem 7.7 Bolzano–Weierstrass ¶
      - depends_on corollary A.45 Archimedean property and density of $\Q$ ¶
      - depends_on corollary A.46 Monotone convergence ¶
      - proves proof ch:05-real-analysis@proof-4 ¶
    - 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 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 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 corollary A.46 Monotone convergence ¶ ↺
      - depends_on theorem A.40 Least-upper-bound property ¶
      - proves proof app:A-long-proofs@proof-34 ¶
    - 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 proposition 7.31 Chain rule ¶
    - depends_on definition 7.13 Composite function ¶
      - depends_on definition 7.9 Real function ¶
    - 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 7.6 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 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 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 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 7.47 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 ¶
- proves proof ch:07-odes-sturm-liouville@proof-73 ¶
proves proof ch:07-odes-sturm-liouville@proof-74 ¶

Supports

depends_on proposition 9.144 The pendulum equation at finite amplitude ¶

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Elliptic integrals of the three kinds	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:5280
`depends_on`	→	Weierstrass $M$-test; uniform limits are continuous	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:5280
`depends_on`	→	Wallis integrals	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:5280
`depends_on`	←	The pendulum equation at finite amplitude	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:5548
`proves`	←	ch:07-odes-sturm-liouville@proof-74	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:5283