std::assoc_...oc_legendrel(3)

💅 std::assoc_legendre, std::assoc_legendref, std::assoc_legendrel ✨

okay bestie so this is from the C++ Standard Library — the std::assoc_...oc_legendrel(3) corner of the manual, and it's literally the page for std::assoc_legendre, std::assoc_legendref, and std::assoc_legendrel. Three icons, one page. Let's get into it 🫶

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🌸 SYNOPSIS 🌸

double      assoc_legendre( unsigned int n, unsigned int m, double x );

double      assoc_legendre( unsigned int n, unsigned int m, float x );
double      assoc_legendre( unsigned int n, unsigned int m, long double x );  (1)
float       assoc_legendref( unsigned int n, unsigned int m, float x );

long double assoc_legendrel( unsigned int n, unsigned int m, long double x );
double      assoc_legendre( unsigned int n, unsigned int m, IntegralType x ); (2)

So like, here's the breakdown bestie:

• (1) — computes the associated Legendre polynomials of the degree n, order m, and argument x. that's the whole main character moment.
• (2) — a whole set of overloads, or a function template, that accepts an argument of any integral type. it's equivalent to (1) after casting the argument to double. cute little disguise, same function underneath 💅

⚠️ okay so important caveat time — fr fr: like all special functions, assoc_legendre is only guaranteed to be available in <cmath> 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. miss one and it's giving "not found" energy 😭

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🎀 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 ✨

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🦋 RETURN VALUE 🦋

If no errors occur, you get the value of the associated Legendre polynomial P^m_n of x, that is

(1 - x²)^(m/2)  ·  dᵐ/dxᵐ P_n(x)

returned — where P_n(x) is the unassociated Legendre polynomial, std::legendre(n, x). yes bestie, it's a whole derivative situation, it's giving calculus main character energy 📐💖

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💀 ERROR HANDLING 💀

Errors may be reported as specified in math_errhandling. ngl, here's the vibe check:

• If the argument is NaN, NaN is returned and domain error is not reported. (silent king behavior ✨)
• If |x| > 1, a domain error may occur. 👀
• If n is greater or equal to 128, the behavior is implementation-defined. so like, don't come for me if it does something weird, okay? 🙈

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🌈 NOTES 🌈

• Implementations that do not support TR 29124 but support TR 19768, provide this function in the header tr1/cmath and namespace std::tr1. retro icon behavior 💅
• An implementation of this function is also available in boost.math. slay, alternate route 🫶
• The first few associated Legendre polynomials are literally:
  • 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).

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⭐ EXAMPLE ⭐

(works as shown with gcc 6.0) — noted, bestie 📝

// Run this code

#define __STDCPP_WANT_MATH_SPEC_FUNCS__ 1
#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

look at that, matching values across the board — we love a consistent queen 👑🔥

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🧚 SEE ALSO 🧚

• legendre — Legendre polynomials
• legendref (function)
• legendrel

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🌸 EXTERNAL LINKS 🌸

Weisstein, Eric W. "Associated Legendre Polynomial." From MathWorld--A Wolfram Web Resource.

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💖 okay that's the whole page bestie, go forth and compute your associated Legendre polynomials responsibly. remember the #define before the includes or it's giving nothing. periodt. 💅

— with love, from cppreference.com, version 2024.06.10 🎀🦋✨