Re: Problem with LegendreQ
- To: mathgroup at smc.vnet.net
- Subject: [mg67617] Re: Problem with LegendreQ
- From: "David W.Cantrell" <DWCantrell at sigmaxi.org>
- Date: Sun, 2 Jul 2006 06:27:48 -0400 (EDT)
- References: <e82o59$r5g$1@smc.vnet.net> <e85ghn$kip$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
"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.) > 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. 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. 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