definition 9.137 Elliptic integrals of the three kinds

open in the book · parts/02-mathematical-methods/07-odes-sturm-liouville.tex:5227 · p. 333

ground object -- no derivation owed

Rests on

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 7.16 Limit ¶
      - depends_on equation 7.7 eq:ana-limit-left ¶
      - depends_on equation 7.5 eq:ana-limit-right ¶
    - depends_on definition 7.4 Convergence ¶
      - depends_on definition 7.2 Absolute value ¶
    - 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 definition A.36 Cut ¶
      - depends_on theorem A.40 Least-upper-bound property ¶
      - proves proof app:A-long-proofs@proof-32 ¶
    - depends_on corollary A.46 Monotone convergence ¶
      - depends_on theorem A.40 Least-upper-bound property ¶ ↺
      - proves proof app:A-long-proofs@proof-33 ¶
    - 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 ¶

Supports

depends_on definition 9.140 Amplitude and the Jacobi elliptic functions ¶
- depends_on proposition 9.143 A cubic quadrature between two turning points ¶
- depends_on proposition 9.141 First properties ¶
  - depends_on proposition 9.143 A cubic quadrature between two turning points ¶ ↺
  - depends_on proposition 9.142 The differential equation of the sine amplitude ¶
- depends_on proposition 9.144 The pendulum equation at finite amplitude ¶
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`	→	Multiple integral	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:5250
`depends_on`	→	Continuous functions are integrable	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:5250
`depends_on`	←	Amplitude and the Jacobi elliptic functions	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:5330
`depends_on`	←	Series for the complete integral of the first kind	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:5280