proposition 23.56 The phase jump at a fold

open in the book · parts/03-classical-mechanics/06-hamilton-jacobi.tex:1904 · p. 845

Rests on

depends_on proposition 23.55 The fold, and the quarter-power law ¶
- depends_on equation 23.86 eq:hj-caustic-jacobian ¶
- depends_on proposition 23.54 The geometrical amplitude diverges ¶
  - depends_on definition 23.52 Lagrangian family; caustic ¶
    - depends_on definition 16.62 Field of extremals; slope function ¶
      - depends_on definition 6.17 Simply connected space ¶
      - depends_on theorem 16.22 Euler–Lagrange ¶
    - depends_on definition 23.5 Complete integral ¶
      - depends_on equation 23.2 eq:hj-equation ¶
  - depends_on equation 23.66 eq:hj-transport ¶
  - depends_on theorem 7.133 Gauss ¶
    - depends_on definition 7.127 Simple regions ¶
      - depends_on definition 7.98 Functions of class $C^{1}$ ¶
      - depends_on definition 6.9 Compact set ¶
    - depends_on remark 7.128 What the derivations below take as given ¶
      - depends_on definition 7.125 Multiple integral ¶
      - depends_on theorem 7.40 Continuous functions are integrable ¶
      - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
    - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
      - depends_on definition 7.41 Antiderivative ¶
      - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
      - proves proof ch:05-real-analysis@proof-25 ¶
    - proves proof ch:05-real-analysis@proof-79 ¶
  - proves proof ch:06-hamilton-jacobi@proof-17 ¶
- proves proof ch:06-hamilton-jacobi@proof-18 ¶
depends_on remark 9.79 Where the one half comes from ¶
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 ¶
    - 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-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 ¶
proves proof ch:06-hamilton-jacobi@proof-19 ¶

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type	direction	node	provenance	where
`depends_on`	→	The fold, and the quarter-power law	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1910
`depends_on`	→	Where the one half comes from	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1910
`depends_on`	→	Connection formulae	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1910
`proves`	←	ch:06-hamilton-jacobi@proof-19	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1913