Re: a bug in Mathematica 7.0?

*To*: mathgroup at smc.vnet.net*Subject*: [mg115950] Re: a bug in Mathematica 7.0?*From*: yaqi <yaqiwang at gmail.com>*Date*: Thu, 27 Jan 2011 03:40:10 -0500 (EST)*References*: <ihh07p$e01$1@smc.vnet.net>

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