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Re: a bug in Mathematica 7.0?

• To: mathgroup at smc.vnet.net
• Subject: [mg115857] Re: a bug in Mathematica 7.0?
• From: Daniel Lichtblau <danl at wolfram.com>
• Date: Sun, 23 Jan 2011 05:34:39 -0500 (EST)

----- Original Message -----
> From: "yaqi" <yaqiwang at gmail.com>
> To: mathgroup at smc.vnet.net
> Sent: Saturday, January 22, 2011 2:22:13 AM
> Subject: [mg115836] 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. What you have is a large matrix, and I do not know which examples are problematic, 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 Support.

Daniel Lichtblau
Wolfram Research