Difference between revisions of "cpp/numeric/special functions/assoc legendre"
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If no errors occur, value of the associated Legendre polynomial {{mathjax-or|1=\(\mathsf{P}_n^m\)|2=P{{su|p=m|b=n}}}} of {{tt|x}}, that is {{mathjax-or|1=\((1 - x^2) ^ {m/2} \: \frac{ \mathsf{d} ^ m}{ \mathsf{d}x ^ m} \, \mathsf{P}_n(x)\)|2=(1-x{{su|p=2}}){{su|p=m/2}} {{mfrac|d{{su|p=m}}|dx{{su|p=m}}}}P{{su|b=n}}(x)}}, is returned (where {{mathjax-or|1=\(\mathsf{P}_n(x)\)|2=P{{su|b=n}}(x)}} is the unassociated Legendre polynomial, {{c|std::legendre(n, x)}}). | If no errors occur, value of the associated Legendre polynomial {{mathjax-or|1=\(\mathsf{P}_n^m\)|2=P{{su|p=m|b=n}}}} of {{tt|x}}, that is {{mathjax-or|1=\((1 - x^2) ^ {m/2} \: \frac{ \mathsf{d} ^ m}{ \mathsf{d}x ^ m} \, \mathsf{P}_n(x)\)|2=(1-x{{su|p=2}}){{su|p=m/2}} {{mfrac|d{{su|p=m}}|dx{{su|p=m}}}}P{{su|b=n}}(x)}}, is returned (where {{mathjax-or|1=\(\mathsf{P}_n(x)\)|2=P{{su|b=n}}(x)}} is the unassociated Legendre polynomial, {{c|std::legendre(n, x)}}). | ||
| − | Note that the [http://mathworld.wolfram.com/Condon-ShortleyPhase.html Condon-Shortley phase term ] {{ | + | Note that the [http://mathworld.wolfram.com/Condon-ShortleyPhase.html Condon-Shortley phase term ] {{mathjax-or|1=\((-1)^m\)|2=(-1){{su|p=m}}}} is omitted from this definition. |
===Error handling=== | ===Error handling=== | ||
Revision as of 18:39, 5 March 2018
| double assoc_legendre( unsigned int n, unsigned int m, double x ); double assoc_legendre( unsigned int n, unsigned int m, float x ); |
(1) | (since C++17) |
| double assoc_legendre( unsigned int n, unsigned int m, Integral x ); |
(2) | (since C++17) |
Contents |
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 |
Return value
If no errors occur, value of the associated Legendre polynomial Pmn ofx, that is (1-x2)m/2 | dm |
| dxm |
Note that the Condon-Shortley phase term (-1)m is omitted from this definition.
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 |x| > 1, a domain error may occur
- If
nis greater or equal to 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 as boost::math::legendre_p, except that the boost.math definition includes the Condon-Shortley phase term.
The first few associated Legendre polynomials are:
- 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) =
(3x2-1)1 2 - assoc_legendre(2, 1, x) = 3x(1-x2)1/2
- assoc_legendre(2, 2, x) = 3(1-x2)
Example
#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
See also
| (C++17)(C++17)(C++17) |
Legendre polynomials (function) |
External links
Weisstein, Eric W. "Associated Legendre Polynomial." From MathWorld--A Wolfram Web Resource.
