definition 9.140 Amplitude and the Jacobi elliptic functions

open in the book · parts/02-mathematical-methods/07-odes-sturm-liouville.tex:5316 · p. 334

ground object -- no derivation owed

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 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 7.9 Real function ¶
      - depends_on definition 3.43 Map ¶
  - 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 ¶
    - depends_on definition 7.2 Absolute value ¶
    - proves proof ch:05-real-analysis@proof-1 ¶
  - 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 ¶

Supports

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 ¶

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`	→	Elliptic integrals of the three kinds	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:5330
`depends_on`	→	Derivative of the inverse function	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:5330
`depends_on`	←	A cubic quadrature between two turning points	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:5464
`depends_on`	←	First properties	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:5359
`depends_on`	←	The pendulum equation at finite amplitude	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:5548