lemma 9.136 Wallis integrals

open in the book · parts/02-mathematical-methods/07-odes-sturm-liouville.tex:5195 · p. 332

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 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 ¶
      - depends_on definition A.36 Cut ¶
      - depends_on definition A.38 Order on $\R$ ¶
      - proves proof app:A-long-proofs@proof-29 ¶
    - 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 ¶
      - 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 ¶
  - 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 corollary A.46 Monotone convergence ¶ ↺
      - depends_on definition 7.45 Series ¶ ↺
      - proves proof ch:05-real-analysis@proof-27 ¶
    - 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 ¶

Supports

depends_on proposition 9.139 Series for the complete integral of the first kind ¶
- depends_on proposition 9.144 The pendulum equation at finite amplitude ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Substitution and integration by parts	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:5201
`depends_on`	→	Derivatives; the Pythagorean identity	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:5201
`depends_on`	←	Series for the complete integral of the first kind	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:5280
`proves`	←	ch:07-odes-sturm-liouville@proof-73	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:5204