proposition 9.103 The positive zeros and their spacing

open in the book · parts/02-mathematical-methods/07-odes-sturm-liouville.tex:4014 · p. 319

Rests on

depends_on corollary 9.96 Recurrence relations ¶
- depends_on proposition 9.95 The two ladder identities ¶
  - depends_on definition 9.93 Bessel functions of the first kind ¶
  - proves proof ch:07-odes-sturm-liouville@proof-45 ¶
- proves proof ch:07-odes-sturm-liouville@proof-46 ¶
depends_on corollary 9.9 Linear equations: existence on the whole interval ¶
- depends_on definition 9.2 Linear equation; homogeneity ¶
  - depends_on definition 9.1 Ordinary differential equation ¶
- depends_on remark 9.3 Normal form and first-order systems ¶
- depends_on theorem 9.8 Picard–Lindelöf ¶
  - depends_on definition 9.4 Lipschitz condition ¶
  - depends_on equation 9.3 eq:slt-ode-system ¶
  - depends_on lemma 9.6 Weierstrass $M$-test; uniform limits are 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 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 ¶
      - proves proof ch:05-real-analysis@proof-28 ¶
    - proves proof ch:07-odes-sturm-liouville@proof-1 ¶
  - proves proof ch:07-odes-sturm-liouville@proof-2 ¶
- proves proof ch:07-odes-sturm-liouville@proof-3 ¶
depends_on proposition 9.102 Liouville normal form of the Bessel equation ¶
- depends_on equation 9.125 eq:slt-bessel-ode ¶
- depends_on remark 9.92 rem:slt-bessel-expanded ¶
- proves proof ch:07-odes-sturm-liouville@proof-50 ¶
proves proof ch:07-odes-sturm-liouville@proof-51 ¶

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type	direction	node	provenance	where
`depends_on`	→	Recurrence relations	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:4025
`depends_on`	→	Linear equations: existence on the whole interval	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:4025
`depends_on`	→	Liouville normal form of the Bessel equation	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:4025
`proves`	←	ch:07-odes-sturm-liouville@proof-51	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:4029