Home > freetb4matlab > specfun > legendre.m

legendre

PURPOSE ^

% Compute the Legendre function of degree @var{n} and order

SYNOPSIS ^

function retval = legendre (n, x, normalization)

DESCRIPTION ^

% -*- texinfo -*-
% @deftypefn {Function File} {@var{l} =} legendre (@var{n}, @var{x})
% @deftypefnx {Function File} {@var{l} =} legendre (@var{n}, @var{x}, @var{normalization})
% Compute the Legendre function of degree @var{n} and order 
% @var{m} = 0 @dots{} N.  The optional argument, @var{normalization}, 
% may be one of @code{'unnorm'}, @code{'sch'}, or @code{'norm'}.
% The default is @code{'unnorm'}.  The value of @var{n} must be a 
% non-negative scalar integer.  
%
% If the optional argument @var{normalization} is missing or is
% @code{'unnorm'}, compute the Legendre function of degree @var{n} and
% order @var{m} and return all values for @var{m} = 0 @dots{} @var{n}.
% The return value has one dimension more than @var{x}.
%
% The Legendre Function of degree @var{n} and order @var{m}:
%
% @example
% @group
%  m        m       2  m/2   d^m
% P(x) = (-1) * (1-x  )    * ----  P (x)
%  n                         dx^m   n
% @end group
% @end example
%
% @noindent
% with Legendre polynomial of degree @var{n}:
%
% @example
% @group
%           1     d^n   2    n
% P (x) = ------ [----(x - 1)  ]
%  n      2^n n~  dx^n
% @end group
% @end example
%
% @noindent
% @code{legendre (3, [-1.0, -0.9, -0.8])} returns the matrix:
%
% @example
% @group
%  x  |   -1.0   |   -0.9   |  -0.8
% ------------------------------------
% m=0 | -1.00000 | -0.47250 | -0.08000
% m=1 |  0.00000 | -1.99420 | -1.98000
% m=2 |  0.00000 | -2.56500 | -4.32000
% m=3 |  0.00000 | -1.24229 | -3.24000 
% @end group
% @end example
%
% If the optional argument @code{normalization} is @code{'sch'}, 
% compute the Schmidt semi-normalized associated Legendre function.
% The Schmidt semi-normalized associated Legendre function is related
% to the unnormalized Legendre functions by the following:
%
% For Legendre functions of degree n and order 0:
%
% @example
% @group
%   0       0
% SP (x) = P (x)
%   n       n
% @end group
% @end example
%
% For Legendre functions of degree n and order m:
%
% @example
% @group
%   m       m          m    2(n-m)~ 0.5
% SP (x) = P (x) * (-1)  * [-------]
%   n       n               (n+m)~
% @end group
% @end example
%
% If the optional argument @var{normalization} is @code{'norm'}, 
% compute the fully normalized associated Legendre function.
% The fully normalized associated Legendre function is related
% to the unnormalized Legendre functions by the following:
%
% For Legendre functions of degree @var{n} and order @var{m}
%
% @example
% @group
%   m       m          m    (n+0.5)(n-m)~ 0.5
% NP (x) = P (x) * (-1)  * [-------------]
%   n       n                   (n+m)~    
% @end group
% @end example
% @end deftypefn

CROSS-REFERENCE INFORMATION ^

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