theorem 9.111 Orthogonality of the Legendre polynomials
open in the book ·
parts/02-mathematical-methods/07-odes-sturm-liouville.tex:4288
· p. 322
Rests on
-
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 ¶
Supports
- depends_on proposition 9.115 Legendre series ¶
-
depends_on
theorem 9.119
Orthogonality at fixed order
¶
- 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 |
→ | Rodrigues' polynomials solve the equation | declared | parts/02-mathematical-methods/07-odes-sturm-liouville.tex:4294 |
depends_on |
→ | Orthogonality of Sturm–Liouville eigenfunctions | declared | parts/02-mathematical-methods/07-odes-sturm-liouville.tex:4294 |
depends_on |
← | Legendre series | declared | parts/02-mathematical-methods/07-odes-sturm-liouville.tex:4468 |
depends_on |
← | Orthogonality at fixed order | declared | parts/02-mathematical-methods/07-odes-sturm-liouville.tex:4593 |
proves |
← | ch:07-odes-sturm-liouville@proof-55 | declared | parts/02-mathematical-methods/07-odes-sturm-liouville.tex:4297 |