associated Legendre function

URNurn:iso:std:iso:80000:-2:ed-2:en:item:2-20.16

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https://metanorma.github.io/isq-smart/math/part-2-20/t2-20.16
urn:iso:std:iso:80000:-2:ed-2:en:item:2-20.16

ISQ Identifiers

ISO

urn:iso:std:iso:80000:-2:ed-2:en:item:2-20.16

IEC

urn:iec:std:iec:80000-2:2019:::item:2-20.16

Designations

ENassociated Legendre function

FRfonction de Legendre associée

Symbol

P_{n}^{m} (z)

Notation systems

AsciiMath

"P"_n^m(z)

LaTeX

\text{P}_{n}^{m} ( z )

Definition

associated Legendre function

fonction de Legendre associée

Notes

$P_{n}^{m} (z) = {(- 1)}^{m} {(1 - z^{2})}^{m / 2} \frac{d^{m}}{d z^{m}} P_{n} (z)$

(for $m, n \in N$ , $m \leq n$ )

Solution of

$(1 - z^{2}) y ″ - 2 z y' + [n (n + 1) - \frac{m^{2}}{1 - z^{2}} \$ y=0]

The factor ${(- 1)}^{m}$ follows from the general theory of spherical functions.

$P_{n}^{m} (z) = {(- 1)}^{m} {(1 - z^{2})}^{m / 2} \frac{d^{m}}{d z^{m}} P_{n} (z)$

(pour $m, n \in N$ , $m \leq n$ )

Solution de

$(1 - z^{2}) y ″ - 2 z y' + [n (n + 1) - \frac{m^{2}}{1 - z^{2}} \$ y=0]

Le facteur ${(- 1)}^{m}$ est déduit de la théorie générale des fonctions sphériques.

Ontology
isq:MathConcept smart:TermEntry

Class Hierarchy

smart:Entity smart:TermEntryisq:MathConcept

A mathematical concept as defined in ISO/IEC 80000 Part 2.

Instance IRI

isq:t2-20.16

Properties

isq:MathConceptShape

Property	Value
rdf:type	isq:MathConcept smart:TermEntry
dcterms:identifier	2-20.16
skosxl:prefLabel	"associated Legendre function"@en
skosxl:altLabel	""P"_n^m(z)"@en
skos:definition	associated Legendre function
skos:note	stem:["P"_n^m(z) = (-1)^m (1-z^2)^(m//2) ("d"^m)/("d"z^m) "P"_n(z)] (for stem:[m, n in bb(sf(N))], stem:[m <= n]) Solu…
smart:hasBindingnessType	bindingness-type:normative
dcterms:isPartOf	isq:part-2-20

Related Entities

isq:MathConceptShape isq:part-2-20 smart:TermEntry