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MathGroup Archive 2005

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Re: LegendreP error (bug?) in Mathematica

  • To: mathgroup at smc.vnet.net
  • Subject: [mg58897] Re: LegendreP error (bug?) in Mathematica
  • From: dh <dh at metrohm.ch>
  • Date: Sat, 23 Jul 2005 05:32:04 -0400 (EDT)
  • References: <dbq2o2$713$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Hi,
Legendre Polynomials are mostly used for n>=0 in LegendreP[n,x]. 
Extending n to negative values does bring nothing new as can be seen 
from Lengendre's diff. eq.:
(1-x^2) y'' - 2x y' + n(n+1) y == 0
replacing n by -n-1 gives the same equation.
However, it is obvious from the above equation that it only determines 
the Legendre polynomial up to a factor. This factor is determined by 
Orthonormality. This still leaves a phase factor undetermined. That is 
why you can encounter different choices mostly +1 or -1.
Obviously, you are using a different choice than mathematica, however, 
this shoud not be a big problem because it is a choice.

sincerely, Daniel

ab at sd.com wrote:
> Why does Mathematica give an erroneous answer to LegendreP[-1,1]=1? where it 
> shoudl give -1 according to the series expansion below? is this a bug?
> 
> In[13]:=
> LegendreP[-1, 1]
> Out[13]=
> 1
> In[12]:=
> p[n_, z_] = (-1)^n*
> Sum[((Pochhammer[-n, k]*Pochhammer[n + 1, k])/k!^2)*
> ((z + 1)/2)^k, {k, 0, n}]
> Out[12]=
> n z + 1
> (-1) Hypergeometric2F1[-n, n + 1, 1, -----]
> 2
> In[16]:=
> p[-1, 1]
> Out[16]=
> -1
> 


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