Timeline for evaluating an integral related to the volume of Hessenberg orthogonal matrices
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| S Jan 20, 2014 at 1:42 | history | suggested | Felix Marin | CC BY-SA 3.0 |
I was fixing several LaTeX items.
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| Jan 20, 2014 at 1:05 | review | Suggested edits | |||
| S Jan 20, 2014 at 1:42 | |||||
| Sep 23, 2010 at 22:17 | vote | accept | John Jiang | ||
| Sep 23, 2010 at 22:14 | history | edited | John Jiang | CC BY-SA 2.5 |
added 10 characters in body
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| Sep 23, 2010 at 4:47 | answer | added | J. M. isn't a mathematician | timeline score: 4 | |
| Sep 23, 2010 at 4:43 | comment | added | John Jiang | You are right. I forgot to divide by $4 \pi^2$, because I was viewing it as an integral over the torus. | |
| Sep 23, 2010 at 3:43 | comment | added | J. M. isn't a mathematician |
Even faster: 12 NIntegrate[EllipticE[(Sin[th/2]/3)^2], {th, 0, 2Pi}]
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| Sep 23, 2010 at 3:28 | comment | added | J. M. isn't a mathematician |
Are you sure about the numerical value you're getting? NIntegrate[Sqrt[9 - Sin[th/2]^2Sin[ph/2]^2], {th, 0, 2Pi}, {ph, 0, 2Pi}] gives 116.7635699899973
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| Sep 23, 2010 at 2:42 | comment | added | John Jiang | It is the volume of the set of orthogonal matrices with a particular form, namely Hessenberg form. It is a natural imbedding of $n-1$ torus in $SO(n)$. Besides that I don't see any natural reason. My guess is I can at most hope to get asymptotics for the volume. | |
| Sep 23, 2010 at 2:21 | comment | added | j.c. | Is there a particular reason you think this integral might have a closed form in special functions or that you need one? In any case you might want to hit the tables of integrals, e.g. Gradshteyn and Ryzhik. | |
| Sep 23, 2010 at 1:52 | comment | added | John Jiang | That's what I found out in mathematica also. Unfortunately mathematica doesn't know what to do next. I was hoping there is some multivariate change of variable that simplies the integral. | |
| Sep 23, 2010 at 1:31 | comment | added | j.c. | I'll point out the obvious (which you may have already tried). You can do one of the $\theta_i$ integrals to get an elliptic integral, probably of the second kind. Not sure what happens next, are there any results on integrals of elliptic integrals? | |
| Sep 23, 2010 at 1:24 | history | asked | John Jiang | CC BY-SA 2.5 |