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

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< cpp‎ | numeric‎ | special math

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 ); 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.

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