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

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< cpp‎ | numeric‎ | special functions

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

1-3) Computes the Associated Legendre polynomials of the degree n, order m, and argument x. The library provides overloads of `std::assoc_legendre` for all cv-unqualified floating-point types as the type of the parameter x.(since C++23)
A) Additional overloads are provided for all integer types, which are treated as double.

### 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

### 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

Pn(x), is returned (where \(\mathsf{P}_n(x)\)Pn(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:

Function | Polynomial
--- | ---
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 )

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

### 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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