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

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Special mathematical 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

double assoc_legendre( unsigned int n, unsigned int m, double x ); double assoc_legendre( unsigned int n, unsigned int m, float x ); double assoc_legendre( unsigned int n, unsigned int m, long double x ); float assoc_legendref( unsigned int n, unsigned int m, float x ); long double assoc_legendrel( unsigned int n, unsigned int m, long double x );	(1)	(since C++17)
double assoc_legendre( unsigned int n, unsigned int m, IntegralType x );	(2)	(since C++17)

1) Computes the associated Legendre polynomials of the degree n, order m, and argument x

2) A set of overloads or a function template accepting an argument of any integral type. Equivalent to (1) after casting the argument to double.

Parameters

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

Return value

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

dm

dxm

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

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

Error handling

Errors may be reported as specified in math_errhandling

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

Notes

Implementations that do not support C++17, but support ISO 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 header tr1/cmath and namespace std::tr1

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

The first few associated Legendre polynomials are:

• assoc_legendre(0, 0, x) = 1
• assoc_legendre(1, 0, x) = x
• assoc_legendre(1, 1, x) = (1-x2
  )1/2
  
• assoc_legendre(2, 0, x) =

  1

  2

  (3x2
  -1)
• assoc_legendre(2, 1, x) = 3x(1-x2
  )1/2
  
• assoc_legendre(2, 2, x) = 3(1-x2
  )

Example

Run this code

#include <cmath>
#include <iostream>
double P20(double x) { return 0.5*(3*x*x-1); }
double P21(double x) { return 3.0*x*std::sqrt(1-x*x); }
double P22(double x) { return 3*(1-x*x); }
int main()
{
    // spot-checks
    std::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';
}

Output:

-0.125=-0.125
1.29904=1.29904
2.25=2.25

See also

legendrelegendreflegendrel (C++17)(C++17)(C++17)

Legendre polynomials (function)

External links

Weisstein, Eric W. "Associated Legendre Polynomial." From MathWorld--A Wolfram Web Resource.