Std::assoc_legendre,std::assoc_legendref,std::assoc_legendrel (3) Linux Manual Page

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

Synopsis

double assoc_legendre(unsigned int n, unsigned int m, double x);

float assoc_legendre(unsigned int n, unsigned int m, float x);

long double assoc_legendre(unsigned int n, unsigned int m, long double x);
(1)(since C++ 17)

    float assoc_legendref(unsigned int n, unsigned int m, float x);

long double assoc_legendrel(unsigned int n, unsigned int m, long double x);

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

External links

Weisstein,_Eric_W._"Associated_Legendre_Polynomial." From MathWorld–A Wolfram Web Resource.

std::assoc_legendre,std::assoc_legendref,std::assoc_legendrel (3) Linux Manual Page