std::assoc_legendre,std::assoc_legendref,std::assoc_legendrel

From cppreference.com

Compiler support
Freestanding and hosted
Language
Standard library
Standard library headers
Named requirements
Feature test macros(C++20)
Language support library
Concepts library(C++20)
Diagnostics library
Memory management library
Metaprogramming library(C++11)
General utilities library
Containers library
Iterators library
Ranges library(C++20)
Algorithms library
Strings library
Text processing library
Numerics library
Date and time library
Input/output library
Filesystem library(C++17)
Concurrency support library(C++11)
Execution control library(C++26)
Technical specifications
Symbols index
External libraries

[edit]

Numerics library

Common mathematical functions

Mathematical special functions(C++17)

Mathematical constants(C++20)

Basic linear algebra algorithms(C++26)

Data-parallel types (SIMD)(C++26)

Floating-point environment(C++11)

Complex numbers

Numeric array (valarray)

Pseudo-random number generation

Bit manipulation(C++20)

Saturation arithmetic(C++26)

(C++17)

(C++17)

(C++20)

(C++20)

Generic numeric operations

iota (C++11)
ranges::iota (C++23)
accumulate
inner_product
adjacent_difference
partial_sum

reduce (C++17)
transform_reduce (C++17)
inclusive_scan (C++17)
exclusive_scan (C++17)
transform_inclusive_scan (C++17)
transform_exclusive_scan (C++17)

C-style checked integer arithmetic

ckd_add (C++26)
ckd_sub (C++26)

ckd_mul (C++26)

[edit]

Mathematical special functions

assoc_laguerreassoc_laguerrefassoc_laguerrel
assoc_legendreassoc_legendrefassoc_legendrel
betabetafbetal
comp_ellint_1comp_ellint_1fcomp_ellint_1l
comp_ellint_2comp_ellint_2fcomp_ellint_2l
comp_ellint_3comp_ellint_3fcomp_ellint_3l
cyl_bessel_icyl_bessel_ifcyl_bessel_il

cyl_bessel_jcyl_bessel_jfcyl_bessel_jl
cyl_bessel_kcyl_bessel_kfcyl_bessel_kl
cyl_neumanncyl_neumannfcyl_neumannl
ellint_1ellint_1fellint_1l
ellint_2ellint_2fellint_2l
ellint_3ellint_3fellint_3l
expintexpintfexpintl

hermitehermitefhermitel
laguerrelaguerreflaguerrel
legendrelegendreflegendrel
riemann_zetariemann_zetafriemann_zetal
sph_besselsph_besselfsph_bessell
sph_legendresph_legendrefsph_legendrel
sph_neumannsph_neumannfsph_neumannl

[edit]

Defined in header`<cmath>`
	(1)
float assoc_legendre(unsignedint n,unsignedint m,float x); double assoc_legendre(unsignedint n,unsignedint m,double x); longdouble assoc_legendre(unsignedint n,unsignedint m,longdouble x);			(since C++17) (until C++23)
/* floating-point-type / assoc_legendre(unsignedint n,unsignedint m, / floating-point-type */ x);			(since C++23)
float assoc_legendref(unsignedint n,unsignedint m,float x);		(2)	(since C++17)
longdouble assoc_legendrel(unsignedint n,unsignedint m,longdouble x);		(3)	(since C++17)
Additional overloads
Defined in header`<cmath>`
template<class Integer> double assoc_legendre(unsignedint n,unsignedint m, Integer x);		(A)	(since C++17)

1-3) Computes theAssociated Legendre polynomials of the degreen, orderm, and argumentx. The library provides overloads ofstd::assoc_legendre for all cv-unqualified floating-point types as the type of the parameterx.(since C++23)

A) Additional overloads are provided for all integer types, which are treated asdouble.

[edit]Parameters

n	-	the degree of the polynomial, an unsigned integer value
m	-	the order of the polynomial, an unsigned integer value
x	-	the argument, a floating-point or integer value

[edit]Return value

If no errors occur, value of the associated Legendre polynomial\(\mathsf{P}_n^m\)Pm
n ofx, that is\((1 - x^2) ^ {m/2} \: \frac{ \mathsf{d} ^ m}{ \mathsf{d}x ^ m} \, \mathsf{P}_n(x)\)(1-x2
)m/2

dxm

Pn(x), is returned (where\(\mathsf{P}_n(x)\)Pn(x) is the unassociated Legendre polynomial,std::legendre(n, x)).

Note that theCondon-Shortley phase term\((-1)^m\)(-1)m
is omitted from this definition.

[edit]Error handling

Errors may be reported as specified inmath_errhandling

If the argument is NaN, NaN is returned and domain error is not reported
If|x| > 1, a domain error may occur
Ifn is greater or equal to 128, the behavior is implementation-defined

[edit]Notes

Implementations that do not support C++17, but supportISO 29124:2010, provide this function if__STDCPP_MATH_SPEC_FUNCS__ is defined by the implementation to a value at least 201003L and if the user defines__STDCPP_WANT_MATH_SPEC_FUNCS__ before including any standard library headers.

Implementations that do not support ISO 29124:2010 but support TR 19768:2007 (TR1), provide this function in the headertr1/cmath and namespacestd::tr1.

An implementation of this function is alsoavailable in boost.math asboost::math::legendre_p, except that the boost.math definition includes the Condon-Shortley phase term.

The first few associated Legendre polynomials are:

Function

Polynomial

assoc_legendre(0,0, x)

assoc_legendre(1,0, x)

assoc_legendre(1,1, x)

(1 - x2
)1/2

assoc_legendre(2,0, x)

(3x2
- 1)

assoc_legendre(2,1, x)

3x(1 - x2
)1/2

assoc_legendre(2,2, x)

3(1 - x2
)

The additional overloads are not required to be provided exactly as(A). They only need to be sufficient to ensure that for their argumentnum of integer type,std::assoc_legendre(int_num1, int_num2, num) has the same effect asstd::assoc_legendre(int_num1, int_num2,static_cast<double>(num)).

[edit]Example

Run this code

#include <cmath>#include <iostream> double P20(double x){return0.5*(3* x* x-1);} double P21(double x){return3.0* x*std::sqrt(1- x* x);} double P22(double x){return3*(1- x* x);} int main(){// spot-checksstd::cout<< std::assoc_legendre(2,0,0.5)<<'='<< P20(0.5)<<'\n'<< std::assoc_legendre(2,1,0.5)<<'='<< P21(0.5)<<'\n'<< std::assoc_legendre(2,2,0.5)<<'='<< P22(0.5)<<'\n';}