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

 

C++

Compiler support

Freestanding and hosted

Language

Standard library headers

Named requirements

Feature test macros (C++20)

Language support library

Concepts library (C++20)

Metaprogramming library (C++11)

Diagnostics library

General utilities library

Strings library

Containers library

Iterators library

Ranges library (C++20)

Algorithms library

Numerics library

Localizations library

Input/output library

Filesystem library (C++17)

Regular expressions library (C++11)

Concurrency support library (C++11)

Technical specifications

Symbols index

External libraries

 

Numerics library

Common mathematical functions

Mathematical special functions (C++17)

Mathematical constants (C++20)

Floating-point environment (C++11)

Complex numbers

Numeric arrays

Pseudo-random number generation

Factor operations

gcd (C++17)

lcm (C++17)

Interpolations

midpoint (C++20)

lerp (C++20)

Generic numeric operations

iota (C++11)

ranges::iota (C++23)

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)

byteswap (C++23)

has_single_bit (C++20)

bit_ceil (C++20)

bit_floor (C++20)

bit_width (C++20)

rotl (C++20)

rotr (C++20)

countl_zero (C++20)

countl_one (C++20)

countr_zero (C++20)

countr_one (C++20)

popcount (C++20)

endian (C++20)

 

Mathematical special 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

 

Defined in header <cmath>
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

See also

assoc_legendreassoc_legendrefassoc_legendrel (C++17)(C++17)(C++17)

associated Legendre polynomials (function)

External links

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