# std::sph_legendre, std::sph_legendref, std::sph_legendrel

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< cpp‎ | numeric‎ | special math

double sph_legendre ( unsigned l, unsigned m, double θ ); float sph_legendre ( unsigned l, unsigned m, float θ ); long double sph_legendre ( unsigned l, unsigned m, long double θ ); float sph_legendref( unsigned l, unsigned m, float θ ); long double sph_legendrel( unsigned l, unsigned m, long double θ ); | (1) | (since C++17)
double sph_legendre ( unsigned l, unsigned m, IntegralType θ ); | (2) | (since C++17)

1) Computes the spherical associated Legendre function of degree `l`, order `m`, and polar angle `θ`.
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

l | - | degree
m | - | order
θ | - | polar angle, measured in radians

### Return value

If no errors occur, returns the value of the spherical associated Legendre function (that is, spherical harmonic with ϕ = 0) of `l`, `m`, and `θ`, where the spherical harmonic function is defined as Ym
l(θ,ϕ) = (-1)m
[

(2l+1)(l-m)!
4π(l+m)!

]1/2
Pm
l(cosθ)eimϕ
where Pm
l(x) is std::assoc_legendre(l,m,x)) and |m|≤l

Note that the Condon-Shortley phase term (-1)m
is included in this definition because it is omitted from the definition of Pm
l in std::assoc_legendre.

### 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 l≥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 the spherical harmonic function is available in boost.math, and it reduces to this function when called with the parameter phi set to zero.

### Example

Run this code

    #include <cmath>
    #include <iostream>
    int main()
    {
        // spot check for l=3, m=0
        double x = 1.2345;
        std::cout << "Y_3^0(" << x << ") = " << std::sph_legendre(3, 0, x) << '\n';

        // exact solution
        double pi = std::acos(-1);
        std::cout << "exact solution = "
                  << 0.25*std::sqrt(7/pi)*(5*std::pow(std::cos(x),3)-3*std::cos(x))
                  << '\n';
    }

Output:

    Y_3^0(1.2345) = -0.302387
    exact solution = -0.302387

### External links

Weisstein, Eric W. "Spherical Harmonic." From MathWorld--A Wolfram Web Resource.

### See also

assoc_legendre assoc_legendref assoc_legendrel (C++17)(C++17)(C++17) | associated Legendre polynomials (function)

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