proposition 9.106 The modified series and what it does not do

open in the book · parts/02-mathematical-methods/07-odes-sturm-liouville.tex:4113 · p. 320

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depends_on definition 9.105 Modified Bessel functions ¶
- depends_on definition 9.93 Bessel functions of the first kind ¶
- depends_on definition 9.54 Sturm–Liouville differential operator ¶
depends_on proposition 9.94 The series solves the Bessel equation ¶
- depends_on definition 9.93 Bessel functions of the first kind ¶ ↺
- depends_on equation 9.125 eq:slt-bessel-ode ¶
- proves proof ch:07-odes-sturm-liouville@proof-44 ¶
depends_on theorem 9.13 Dimension of the solution space ¶
- 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 proposition 7.47 Comparison; absolute convergence ¶
      - 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 definition 9.2 Linear equation; homogeneity ¶ ↺
- proves proof ch:07-odes-sturm-liouville@proof-6 ¶
proves proof ch:07-odes-sturm-liouville@proof-52 ¶

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type	direction	node	provenance	where
`depends_on`	→	Modified Bessel functions	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:4122
`depends_on`	→	The series solves the Bessel equation	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:4122
`depends_on`	→	Dimension of the solution space	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:4122
`proves`	←	ch:07-odes-sturm-liouville@proof-52	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:4126