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Re: Problem with LegendreQ

  • To: mathgroup at smc.vnet.net
  • Subject: [mg67618] Re: Problem with LegendreQ
  • From: "sashap" <pavlyk at gmail.com>
  • Date: Sun, 2 Jul 2006 06:27:54 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

On 7/1/06, David W. Cantrell <DWCantrell at sigmaxi.org> wrote:
> [Message also posted to: comp.soft-sys.math.mathematica]
>
> "sashap" <pavlyk at gmail.com> wrote:
> > Hi Michael,
> >
> > There is no problem with Mathematica's LegendreQ. In fact there is a
> > typo is A&S.
>
> No. In fact, there is no typo about that in A&S. (Yes, there are still
> mistakes even in the last printing of A&S, but that's not one of them.)

By the typo, I refer to the inconsistency between AS 17.3.9 (which is
consistent
with Mathematica's definition of EllipticK) and AS 8.13.3 (where
apparently a
different definition of EllipticK is used). This is a typo,
considering the book as
a whole.

>
> > The correct identity reads:
> >
> > LegendreQ[-1/2, 0, 3, x] = Sqrt[2/(x+1)] * EllipticK[ 2/(x+1) ]
>
> That seems to be valid throughout the complex plane, except on the real
> axis for x <= -1.

Correct, but my reply should be put in the context of Michael's
question.
He refers to AS 8.13.3 stated there for Re[x] > 0, so my reply assumed
that
restriction as well.

>
> The above identity is expressed using Mathematica's notational convention
> for elliptic integrals. The corresponding identity in A&S is also correct;
> it merely _looks_ different due to the fact that A&S use a different (and
> also very common) notational convention for elliptic integrals. [I don't
> know Michael's background. If he's not an experienced user of elliptic
> integrals, it's understandable that he would be confused by the differing
> conventions. But I would have certainly thought that any Special Functions
> Developer would be excruciatingly well aware of the differing conventions!]
>
> > Notice that LegendreQ[-1/2,x] is a short form of
> >
> > LegendreQ[-1/2, 0, 2, x], that is LegendreQ[-1/2, x] has different
> > branch-cut structure than LegendreQ[-1/2, 0, 3, x].
> >
> > The typo in A&S becomes apparent if you compare
> >
> > In[2]:= LegendreP[-2^(-1), x]
> >
> > Out[2]= (2*EllipticK[(1 - x)/2])/Pi
> >
> > to AS which incorrectly gives
> >   LegendreP[-1/2,x] == (2*EllipticK[ Sqrt[ (1 - x)/2 ]  ])/Pi.
>
> Again, what Mathematica gives and what A&S give are equivalent. They are
> merely expressed using two different common conventions for notating
> elliptic integrals.

My remark from above applies here as well.

Oleksandr Pavlyk
Special Functions Developer
Wolfram Research

>
> David W. Cantrell
>
>
> > Oleksandr Pavlyk,
> > Special Functions Developer
> > Wolfram Research
> >
> > mmandelberg at comcast.net wrote:
> > > I seem to be getting incorrect numerical results for the LegendreQ[n,x]
> > > function.  For example I get:
> > >
> > > LegendreQ[-1/2,5]//N     =    1.00108 -1.17142 i
> > >
> > > so that the imaginary part is not zero.  However, using the identity
> > > (Abramowitz and Stegun 8.3.13):
> > >
> > > LegendreQ[-1/2, x] = Sqrt[2/(x+1)] * EllipticK[Sqrt[2/(x+1)]]
> > >
> > > The answer should be:
> > >
> > > Sqrt[2/(5+1)]EllipticK[Sqrt[2/(5+1)]]//N
> > >
> > >  1.11187
> > >
> > >
> > > Michael Mandelberg
>


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