lemma 9.56 Lagrange identity

open in the book · parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1913 · p. 294

Rests on

depends_on definition 9.54 Sturm–Liouville differential operator ¶
proves proof ch:07-odes-sturm-liouville@proof-28 ¶

Supports

depends_on lemma A.442 The Wronskian of the accessory equation is constant ¶
- depends_on definition A.467 Green function of the shifted problem ¶
  - depends_on lemma A.468 The kernel is well defined, symmetric and Lipschitz ¶
    - depends_on theorem A.470 $T$ is compact ¶
      - depends_on theorem A.471 Spectral decomposition and completeness in $L^{2}_{r}$ ¶
  - depends_on theorem A.469 The Green operator inverts $L_{K}$ ¶
    - depends_on lemma A.472 The pairing identity ¶
    - depends_on theorem A.471 Spectral decomposition and completeness in $L^{2}_{r}$ ¶ ↺
- depends_on theorem A.443 Sturm separation theorem ¶
depends_on theorem 9.57 Orthogonality of Sturm–Liouville eigenfunctions ¶
- depends_on phenomenon 28.53 Chladni's figures ¶
- depends_on phenomenon 28.45 The harmonic series of a stretched string ¶
- depends_on theorem 9.119 Orthogonality at fixed order ¶
  - depends_on theorem 9.122 Normalisation and orthonormality ¶
- depends_on theorem 9.98 Orthogonality of the Bessel functions ¶
  - depends_on proposition 9.100 Fourier–Bessel coefficients ¶
- depends_on theorem 9.127 Orthogonality ¶
  - depends_on proposition 9.128 Hermite series ¶
- depends_on theorem 9.133 Orthonormality ¶
  - depends_on proposition 9.134 Laguerre series ¶
- depends_on theorem 9.111 Orthogonality of the Legendre polynomials ¶
  - depends_on proposition 9.115 Legendre series ¶
  - depends_on theorem 9.119 Orthogonality at fixed order ¶ ↺

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`	→	Sturm–Liouville differential operator	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1921
`depends_on`	←	The Wronskian of the accessory equation is constant	declared	appendices/A-long-proofs.tex:21748
`depends_on`	←	Orthogonality of Sturm–Liouville eigenfunctions	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1952
`proves`	←	ch:07-odes-sturm-liouville@proof-28	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1924