Re: a bug in Mathematica 7.0?
- To: mathgroup at smc.vnet.net
- Subject: [mg115979] Re: a bug in Mathematica 7.0?
- From: Yaqi Wang <yaqiwang at gmail.com>
- Date: Fri, 28 Jan 2011 06:13:25 -0500 (EST)
Tony, Sorry about my notation. Here mu is the polor angle and tht is the azimuthal angle. Forget my original post. It is too long although it is correct and maybe useful. The problem is identified in my last previous post, which is much shorter. Best, Yaqi On Thu, Jan 27, 2011 at 7:57 AM, Tony Harker <a.harker at ucl.ac.uk> wrote: > Yaqi, > Surely your real spherical harmonics should involve Sin[2 mu] and > Sin[4 mu], not Sin[mu]^2 and Sin[mu]^4? You got away with it for your > version of (l=2, m=2) and (l=4, m=4) because the theta parts integrate to > zero, but that doesn't work for the (l=4,m=2) (l=4, m=4) case. It's obvious > that the mu parts cannot integrate to zero, as they are both non-negative > for all real mu. > > Tony Harker > > > ]-> -----Original Message----- > ]-> From: yaqi [mailto:yaqiwang at gmail.com] > ]-> Sent: 27 January 2011 08:40 > ]-> To: mathgroup at smc.vnet.net > ]-> Subject: [mg115950] Re: a bug in Mathematica 7.0? > ]-> > ]-> On Jan 23, 3:34 am, Daniel Lichtblau <d... at wolfram.com> wrote: > ]-> > ----- Original Message ----- > ]-> > > From: "yaqi" <yaqiw... at gmail.com> > ]-> > > To: mathgr... at smc.vnet.net > ]-> > > Sent: Saturday, January 22, 2011 2:22:13 AM > ]-> > > Subject: a bug in Mathematica 7.0? > ]-> > > Hello, > ]-> > > ]-> > > I was shocked by the integration result of spherical harmonics > given > ]-> > > by Mathematica 7.0. The notebook conducting these evaluations is > ]-> > > attached at the end of this post. > ]-> > > ]-> > > Basically, I create a vector of real harmonics Y={Y_{n,k},k=- > ]-> > > n,n;n=0,4} and then integrate Y_{n,k}*Y_{n,k}*Omega_y over the > ]-> > > entire 2D sphere. The integral should be zero for > ]-> > > Y_{2,2}*Y_{4,-4}*Omega_y but Mathematica 7.0 gives me > ]-> > > -55*Sqrt[21]/512. Similar for Y_{4,2} *Y_{4,-4}*Omega_y, it should > be > ]-> zero but I get 99*Sqrt[7]/2048. > ]-> > > ]-> > > So I create another vector of normal spherical harmonics by using > ]-> > > 'SphericalHarmonicY' and then map it to the real harmonics and do > ]-> > > the integral mentioned above. The only difference is that I have a > ]-> > > change of variable in this integral; instead of using the cosine of > ]-> > > the polor angle, I used the polor angle for the intergal directly. > ]-> > > This time, Mathematica 7.0 gives me correct results. > ]-> > > ]-> > > The only different between the two results are the two terms I > ]-> > > mentioned above. I did the similar thing with Mathematica 5.0. > ]-> > > Everything is correct. > ]-> > > ]-> > > So can somebody take a look on the notebook, see if I messed up > ]-> some > ]-> > > variable usages or this is indeed a bug in Mathematica 7.0? I use > ]-> > > Mathematica 7.0 for my regular derivations, this really shocked me! > ]-> > > ]-> > > I do not know how to attach a file, so I copy and paste the entire > ]-> > > notebook and attached below. > ]-> > > ]-> > > Many thanks. > ]-> > > [...] > ]-> > > ]-> > Please send the integrand and expected result for one of the bad > ]-> > cases. W= > ]-> hat you have is a large matrix, and I do not know which examples are > ]-> proble= matic, let alone what specific integrands produced them. (For > ]-> example, I do= now know what integrand goes with the statement > ]-> "Y_{2,2}*Y_{4,-4}*Omega_y"= . Maybe this is inexcusable ignorance on my > ]-> part. Humor me.) > ]-> > > ]-> > Can send to any or all of myself, MathGroup, or Wolfram Research Tech > ]-> > Sup= > ]-> port. > ]-> > > ]-> > Daniel Lichtblau > ]-> > Wolfram Research- Hide quoted text - > ]-> > > ]-> > - Show quoted text - > ]-> > ]-> Sorry for the long original post. I separated the problem below: > ]-> > ]-> In[1]:= 1/4 Sqrt[15/\[Pi]] Cos[2 tht] Sin[mu]^2 > ]-> > ]-> Out[1]= 1/4 Sqrt[15/\[Pi]] Cos[2 tht] Sin[mu]^2 > ]-> > ]-> In[2]:= 3/16 Sqrt[35/\[Pi]] Sin[mu]^4 Sin[4 tht] > ]-> > ]-> Out[2]= 3/16 Sqrt[35/\[Pi]] Sin[mu]^4 Sin[4 tht] > ]-> > ]-> In[3]:= 1/8 E^(-2 I tht) (1 + E^(4 I tht)) Sqrt[15/\[Pi]] Sin[mu]^2 > ]-> > ]-> Out[3]= 1/8 E^(-2 I tht) (1 + E^(4 I tht)) Sqrt[15/\[Pi]] Sin[mu]^2 > ]-> > ]-> In[4]:= -(3/32) I E^(-4 I tht) (-1 + E^(8 I tht)) Sqrt[35/\[Pi]] > ]-> Sin[mu]^4 > ]-> > ]-> Out[4]= -(3/32) I E^(-4 I tht) (-1 + E^(8 I tht)) Sqrt[35/\[Pi]] > ]-> Sin[mu]^4 > ]-> > ]-> In[5]:= Simplify[%3 - %1] > ]-> > ]-> Out[5]= 0 > ]-> > ]-> In[6]:= Simplify[%4 - %2] > ]-> > ]-> Out[6]= 0 > ]-> > ]-> In[7]:= Integrate[%1*%2*(-Sin[mu]*Cos[tht])*Sin[mu], {mu, 0, > ]-> Pi}, {tht, 0, 2*Pi}] > ]-> > ]-> Out[7]= -((55 Sqrt[21])/512) > ]-> > ]-> In[8]:= Integrate[%3*%4*(-Sin[mu]*Cos[tht])*Sin[mu], {mu, 0, > ]-> Pi}, {tht, 0, 2*Pi}] > ]-> > ]-> Out[8]= 0 > ]-> > ]-> > ]-> ======================= > ]-> Out[7] and Out[8] should be the same, but they are not. This is really > ]-> troubling me. Am I able to get a patch after it is fixed? > ]-> > ]-> Thanks. > ]-> > ]-> Yaqi > >