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std::legendre, std::legendref, std::legendrel

double      legendre( unsigned int n, double x );
float       legendre( unsigned int n, float x );
long double legendre( unsigned int n, long double x );
float       legendref( unsigned int n, float x );
long double legendrel( unsigned int n, long double x );

(1)

(since C++17)

double      legendre( unsigned int n, IntegralType x );

(2)

(since C++17)

1) Computes the unassociated Legendre polynomials of the degree n and argument x

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

n	-	the degree of the polynomial
x	-	the argument, a value of a floating-point or integral type

Return value

If no errors occur, value of the order-n unassociated Legendre polynomial of x, that is \(\mathsf{P}_n(x) = \frac{1}{2^n n!} \frac{\mathsf{d}^n}{\mathsf{d}x^n} (x^2-1)^n \)

2n
n!

dxn

(x2
-1)n
, is returned.

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
The function is not required to be defined for |x|>1
If n is greater or equal than 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.

The first few Legendre polynomials are:

legendre(0, x) = 1
legendre(1, x) = x
legendre(2, x) =

1

2

(3x2
-1)
legendre(3, x) =

1

2

(5x3
-3x)
legendre(4, x) =

1

8

(35x4
-30x2
+3)

Example

#include <cmath>
#include <iostream>
double P3(double x) { return 0.5*(5*std::pow(x,3) - 3*x); }
double P4(double x) { return 0.125*(35*std::pow(x,4)-30*x*x+3); }
int main()
{
    // spot-checks
    std::cout << std::legendre(3, 0.25) << '=' << P3(0.25) << '\n'
              << std::legendre(4, 0.25) << '=' << P4(0.25) << '\n';
}

Output:

-0.335938=-0.335938
0.157715=0.157715

External links

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

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http://en.cppreference.com/w/cpp/numeric/special_math/legendre

laguerrelaguerreflaguerrel (C++17)(C++17)(C++17)	Laguerre polynomials (function)
hermitehermitefhermitel (C++17)(C++17)(C++17)	Hermite polynomials (function)