std::legendre, std::legendref, std::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      legendre( unsigned int n, double x ); float       legendre( unsigned int n, float x ); long double legendre( unsigned int n, long double x ); float       legendref( unsigned int n, float x ); long double legendrel( unsigned int n, long double x );	(1)	(since C++17)
double      legendre( unsigned int n, IntegralType x );	(2)	(since C++17)

1) Computes the unassociated Legendre polynomials of the degree n 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
x	-	the argument, a value of a floating-point or integral type

Return value

If no errors occur, value of the order-n unassociated Legendre polynomial of x, that is \(\mathsf{P}_n(x) = \frac{1}{2^n n!} \frac{\mathsf{d}^n}{\mathsf{d}x^n} (x^2-1)^n \)

1

2n n!

dn

dxn

(x2
-1)n
, is returned.

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
• The function is not required to be defined for |x|>1
• If n is greater or equal than 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

The first few Legendre polynomials are:

• legendre(0, x) = 1
• legendre(1, x) = x
• legendre(2, x) =

  1

  2

  (3x2
  -1)
• legendre(3, x) =

  1

  2

  (5x3
  -3x)
• legendre(4, x) =

  1

  8

  (35x4
  -30x2
  +3)

Example

Run this code

#include <cmath>
#include <iostream>
double P3(double x) { return 0.5*(5*std::pow(x,3) - 3*x); }
double P4(double x) { return 0.125*(35*std::pow(x,4)-30*x*x+3); }
int main()
{
    // spot-checks
    std::cout << std::legendre(3, 0.25) << '=' << P3(0.25) << '\n'
              << std::legendre(4, 0.25) << '=' << P4(0.25) << '\n';
}

Output:

-0.335938=-0.335938
0.157715=0.157715

See also

laguerrelaguerreflaguerrel (C++17)(C++17)(C++17)	Laguerre polynomials (function)
hermitehermitefhermitel (C++17)(C++17)(C++17)	Hermite polynomials (function)

External links

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