Re: Problem with LegendreQ
- To: mathgroup at smc.vnet.net
- Subject: [mg67592] Re: Problem with LegendreQ
- From: "sashap" <pavlyk at gmail.com>
- Date: Sat, 1 Jul 2006 05:12:41 -0400 (EDT)
- References: <e82o59$r5g$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Hi Michael, There is no problem with Mathematica's LegendreQ. In fact there is a typo is A&S. The correct identity reads: LegendreQ[-1/2, 0, 3, x] = Sqrt[2/(x+1)] * EllipticK[ 2/(x+1) ] 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. 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