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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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