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

 

C++

Language

Standard Library Headers

Freestanding and hosted implementations

Named requirements

Language support library

Concepts library (C++20)

Diagnostics library

Utilities library

Strings library

Containers library

Iterators library

Ranges library (C++20)

Algorithms library

Numerics library

Input/output library

Localizations library

Regular expressions library (C++11)

Atomic operations library (C++11)

Thread support library (C++11)

Filesystem library (C++17)

Technical Specifications

 

Numerics library

Common mathematical functions

Special mathematical functions

Floating-point environment (C++11)

Complex numbers

Numeric arrays

Pseudo-random number generation

Compile-time rational arithmetic (C++11)

Numeric algorithms

gcd (C++17)

lcm (C++17)

Generic numeric operations

iota (C++11)

accumulate

inner_product

adjacent_difference

partial_sum

reduce (C++17)

transform_reduce (C++17)

inclusive_scan (C++17)

exclusive_scan (C++17)

transform_inclusive_scan (C++17)

transform_exclusive_scan (C++17)

Bit operations

bit_cast (C++20)

ispow2 (C++20)

ceil2 (C++20)

floor2 (C++20)

log2p1 (C++20)

 

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      sph_legendre ( unsigned l, unsigned m, double θ ); double      sph_legendre ( unsigned l, unsigned m, float θ ); 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_legendreassoc_legendrefassoc_legendrel (C++17)(C++17)(C++17)

associated Legendre polynomials (function)