proposition 23.57 The wave resolution at a turning surface

open in the book · parts/03-classical-mechanics/06-hamilton-jacobi.tex:1933 · p. 845

Rests on

depends_on remark 23.51 Where the expansion breaks, and what is owed ¶
- depends_on proposition 9.65 Validity criterion ¶
  - depends_on definition 9.64 Reduced de Broglie wavelength ¶
    - depends_on equation 9.60 eq:slt-wkb-normal-form ¶
  - depends_on equation 9.64 eq:slt-wkb-series ¶
  - depends_on proposition 9.61 Eikonal and transport equations ¶
    - depends_on equation 9.64 eq:slt-wkb-series ¶ ↺
    - depends_on proposition 9.60 Exact Riccati form ¶
      - depends_on equation 9.62 eq:slt-wkb-ansatz ¶
      - depends_on equation 9.60 eq:slt-wkb-normal-form ¶ ↺
      - proves proof ch:07-odes-sturm-liouville@proof-30 ¶
    - proves proof ch:07-odes-sturm-liouville@proof-31 ¶
  - proves proof ch:07-odes-sturm-liouville@proof-34 ¶
- depends_on theorem 9.72 Connection formulae ¶
  - depends_on corollary 9.62 The semiclassical wave ¶
    - depends_on equation 9.62 eq:slt-wkb-ansatz ¶ ↺
    - depends_on equation 9.61 eq:slt-wkb-kappa ¶
    - depends_on proposition 9.61 Eikonal and transport equations ¶ ↺
    - proves proof ch:07-odes-sturm-liouville@proof-32 ¶
  - depends_on equation 9.79 eq:slt-wkb-airy-ai ¶
  - depends_on equation 9.80 eq:slt-wkb-airy-bi ¶
  - depends_on equation 9.76 eq:slt-wkb-airy-equation ¶
  - depends_on equation 9.74 eq:slt-wkb-airy-scale ¶
  - proves proof ch:07-odes-sturm-liouville@proof-37 ¶
depends_on theorem 9.70 Asymptotics of the Airy functions ¶
- depends_on definition 9.68 Airy functions ¶
- depends_on proposition 9.69 The integral solves the Airy equation ¶
  - depends_on definition 9.68 Airy functions ¶ ↺
  - depends_on equation 9.76 eq:slt-wkb-airy-equation ¶ ↺
  - proves proof ch:07-odes-sturm-liouville@proof-35 ¶
- proves proof ch:07-odes-sturm-liouville@proof-36 ¶
depends_on theorem 9.72 Connection formulae ¶ ↺
proves proof ch:06-hamilton-jacobi@proof-20 ¶

Supports

depends_on remark 23.58 The caustics that are not turning surfaces ¶
depends_on remark 23.59 What is observed ¶

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`	→	Where the expansion breaks, and what is owed	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1945
`depends_on`	→	Asymptotics of the Airy functions	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1945
`depends_on`	→	Connection formulae	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1945
`depends_on`	←	The caustics that are not turning surfaces	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1987
`depends_on`	←	What is observed	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:2096
`proves`	←	ch:06-hamilton-jacobi@proof-20	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1949