◆ legendre

kyosu::legendre = eve::functor<legendre_t>

inlineconstexpr

Computes the value of the Legendre and associated Legendre functions of order n ( and 'm) atx`:

For positive integer n the standard legendre functions are polynomials:

The Legendre polynomial of order n is given by \(\displaystyle \mbox{L}_{n}(x) = \frac{e^x}{n!}\frac{d^n}{dx^n}(x^ne^{-x})\).

However the function legendre must and can be called with real or complex parameters.

Header file

#include <kyosu/functions.hpp>

Callable Signatures

namespace eve
{
   // Regular overload
   auto constexpr legendre(kyosu::concepts::real auto n, kyosu::concepts::cayley_dickson auto z) noexcept; // 1
   auto constexpr legendre(kyosu::concepts::real auto n, kyosu::concepts::real auto z)           noexcept; // 1
 
   // Lanes masking
   constexpr auto legendre[conditional_expr auto c](/* any previous overload */)                 noexcept; // 2
   constexpr auto legendre[logical_value auto m](/* any previous overload */)                    noexcept; // 2
 
   // Semantic options
   constexpr auto legendre[successor](integral_value auto n,
                                      kyosu::concepts::cayley_dickson auto z,
                                      kyosu::concepts::cayley_dickson auto pn,
                                      kyosu::concepts::cayley_dickson auto pnm1)                  noexcept; // 3
}

Parameters

n : integral positive arguments.
z : cayley_dickson or real argument
pn, pnm1 : cayley_dickson arguments
c: Conditional expression masking the operation.
m: Logical value masking the operation.

Return value

The value of the Legendre polynomial of order n at z is returned.
The operation is performed conditionnaly.
The successor option implements the three term recurrence relation for the (associated) Legendre functions, \(\displaystyle \mbox{P}_{l+1} = \left((2l+1)\mbox{P}_{l}(x)-l\mbox{P}_{l-1}(x)\right)/(l+1)\).

◆ legendre

Header file

Callable Signatures

External references

Example

	kyosu v0.1.0 Complex Without Complexes