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Re: LegendreP (Symbolic) is different in Mathematica5 than previous versions (M4, M3 ..)

• To: mathgroup at smc.vnet.net
• Subject: [mg51877] Re: LegendreP (Symbolic) is different in Mathematica5 than previous versions (M4, M3 ..)
• From: Paul Abbott <paul at physics.uwa.edu.au>
• Date: Thu, 4 Nov 2004 01:49:28 -0500 (EST)
• Organization: The University of Western Australia
• References: <200410160820.EAA23725@smc.vnet.net> <DCE69724-1F76-11D9-9EF0-000A95B4967A@mimuw.edu.pl> <cl2fa3$mp1$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

In article <cl2fa3$mp1$1 at smc.vnet.net>,
 "Peter S Aptaker" <psa at laplacian.co.uk> wrote:

> Thanks. As with much or most use of LegendreP the argument  mu is real.  My
> point and concern is in the Subject:

Real -- but over what range? And I disagree that the argument z (mu is 
usually used as one of the parameters) is usually real, at least in the 
applications I need it for. I assume that you are restricting attention 
to z = x in -1 < x < 1. In that case, Andrzej Kozlowski's suggestion of 
using 

  P[n_, m_, x_] := (-1)^m (1 - x^2)^(m/2) D[LegendreP[n, x], x]

(for m >= 0) is one way to go.

Remember, Mathematica is designed to work with functions (and parameters 
and arguments) assumed to be complex, in general.

> "LegendreP (Symbolic) is different in Mathematica 5 than previous versions
> (M4, M3 ..)"
> 
> Your comment means ttht the Mathematica 5 result is not simply 'a different 
> form'  but 'the wrong answer'?

The answer is _not_ wrong. If you look at the online documentation for 
LegendreP you will see that it accepts FOUR arguments. By default 
LegendreP[1,1,mu] gives the "Type 1" case, which is defined only for z 
within the unit circle in the complex plane. Indeed, although

 FullSimplify[Sqrt[(-z - 1)/(z - 1)] (z - 1) + Sqrt[1 - z^2], Abs[z] < 1]

returns 0, FullSimplify fails to reduce Sqrt[(-z - 1)/(z - 1)] (z - 1) 
to -Sqrt[1 - z^2]. But the result is not wrong. The definition needs to 
handle different possible branch cuts. For example, Type 3 functions 
have a single branch cut from -Infinity to +1. 

Cheers,
Paul 

> ----- Original Message ----- 
> From: "Andrzej Kozlowski" <akoz at mimuw.edu.pl>
To: mathgroup at smc.vnet.net
> Subject: [mg51877] Re:  LegendreP (Symbolic) is different in Mathematica5
> than previous versions (M4, M3 ..)
> 
> 
> > On 16 Oct 2004, at 17:20, peteraptaker wrote:
> >
> >> LegendreP[1,1,mu]
> >> Out[with m4] = -Sqrt[1 - mu^2]
> >> Out[with m5] = Sqrt[(-1 - mu)/(-1 + mu)]*(-1 + mu)
> >>
> >> While this ( and common sense) show they are equal ..
> >>
> >> dum = m4 - m5 // FullSimplify
> >> PowerExpand[dum]
> >> Out[]= 0
> >>
> >
> > Well, it seems to me that the commonsense thing to do in such situations
> > is not to rely too much on common sense and even less on PowerExpand. In
> > fact your two expressions are certainly not equal:
> >
> > a[mu_] := -Sqrt[1 - mu^2];
> > b[mu_] := Sqrt[(-1 - mu)/(-1 + mu)]*(-1 + mu);
> >
> > a[2]
> > (-I)*Sqrt[3]
> >
> > FullSimplify[b[2]]
> >
> > I*Sqrt[3]
> >
> >
> > Andrzej Kozlowski
> > Chiba, Japan
> > http://www.akikoz.net/~andrzej/
> > http://www.mimuw.edu.pl/~akoz/
> >
> >
> 
>

-- 
Paul Abbott                                   Phone: +61 8 6488 2734
School of Physics, M013                         Fax: +61 8 6488 1014
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