Timeline for Application of Feynman parameters in an improper integral
Current License: CC BY-SA 4.0
9 events
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| 9 hours ago | comment | added | guest | Carlo, I found that you can get to the equation in the OP from your final expression if you make the change of variables: $u_1=t$, $u_2=s\cdot(1-t)$, and $u_3=(1-s)\cdot(1-t)$. This has the added benefit that the integration over the simplex spanned by $u_2$, $u_3$ is now transformed into an integration over $[0,1]\times[0,1]$ | |
| Oct 1 at 13:14 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
symmetric order
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| Oct 1 at 2:14 | comment | added | guest | Thank you! Exactly what I was looking for, and I've accepted the answer, I really appreciate the time spent on it. Removing the w dependence (associated to the so-called "off-knot" points), helps a lot with numerical applications, and one can go further and integrate over u_2 and u_3. ...in a few days i'll follow-up and add more to the original post, just to show the additional steps of integrating over the remaining Feynman parameters given this answer...which I understand is possible also from the linked paper (but the steps are easier :) ) | |
| Oct 1 at 1:57 | vote | accept | guest | ||
| Sep 30 at 21:24 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
strike through the symmetry statement
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| Sep 30 at 14:03 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
added 4 characters in body
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| Sep 30 at 13:22 | comment | added | Michael Engelhardt | The parametrization in the OP is also clearly symmetric in $a$ and $b$, one just has to couple the exchange of $a$ and $b$ with $s\rightarrow 1-s$. Whereas in your result, it's coupled with exchanging two variables, $u_2 $ and $u_3 $. In that sense, the structure is quite different. Interesting that one can use Feynman's trick to go into diverging directions here. | |
| Sep 30 at 13:02 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
corrected typo
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| Sep 30 at 12:18 | history | answered | Carlo Beenakker | CC BY-SA 4.0 |