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 >