theorem 9.119 Orthogonality at fixed order

open in the book · parts/02-mathematical-methods/07-odes-sturm-liouville.tex:4585 · p. 326

Rests on

depends_on proposition 9.118 The construction solves the associated equation ¶
- depends_on definition 9.116 Associated Legendre functions ¶
  - depends_on definition 9.108 Rodrigues formula ¶
- depends_on lemma 9.117 The differentiated Legendre equation ¶
  - depends_on equation 9.146 eq:slt-legendre-ode ¶
  - proves proof ch:07-odes-sturm-liouville@proof-59 ¶
- proves proof ch:07-odes-sturm-liouville@proof-60 ¶
depends_on theorem 9.111 Orthogonality of the Legendre polynomials ¶
- depends_on proposition 9.109 Rodrigues' polynomials solve the equation ¶
  - depends_on definition 9.108 Rodrigues formula ¶ ↺
  - depends_on equation 9.146 eq:slt-legendre-ode ¶ ↺
  - proves proof ch:07-odes-sturm-liouville@proof-53 ¶
- depends_on theorem 9.57 Orthogonality of Sturm–Liouville eigenfunctions ¶
  - depends_on equation 9.55 eq:slt-eigenproblem ¶
  - depends_on lemma 9.56 Lagrange identity ¶
    - depends_on definition 9.54 Sturm–Liouville differential operator ¶
    - proves proof ch:07-odes-sturm-liouville@proof-28 ¶
  - proves proof ch:07-odes-sturm-liouville@proof-29 ¶
- proves proof ch:07-odes-sturm-liouville@proof-55 ¶
depends_on theorem 9.57 Orthogonality of Sturm–Liouville eigenfunctions ¶ ↺
proves proof ch:07-odes-sturm-liouville@proof-61 ¶

Supports

depends_on theorem 9.122 Normalisation and orthonormality ¶

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`	→	The construction solves the associated equation	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:4593
`depends_on`	→	Orthogonality of the Legendre polynomials	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:4593
`depends_on`	→	Orthogonality of Sturm–Liouville eigenfunctions	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:4593
`depends_on`	←	Normalisation and orthonormality	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:4745
`proves`	←	ch:07-odes-sturm-liouville@proof-61	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:4600