Re: a bug in Mathematica 7.0?
- To: mathgroup at smc.vnet.net
- Subject: [mg116002] Re: a bug in Mathematica 7.0?
- From: "Tony Harker" <a.harker at ucl.ac.uk>
- Date: Fri, 28 Jan 2011 06:17:48 -0500 (EST)
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