I was just rummaging through some old files and folders when I found some bizarre attempts at deriving some sort of (I think) probability density functions. I don’t think they ended up being useful for my thesis, but they are really quite an impressive output from my computer algebra system.
\[\left( z \mapsto \frac{\int\limits_{-\infty}^{\infty} e^{- \frac{x_{1}^{2}}{2 s_{1}^{2}}} \int\limits_{-\infty}^{\infty} e^{- \frac{x_{0}^{2}}{2 s_{0}^{2}}} \delta\left(- z + \frac{x_{0} x_{1}}{\left|{x_{0}}\right| \left|{x_{1}}\right|}\right)\, dx_{0}\, dx_{1}}{2 \pi s_{0} s_{1}} \right)\]
\[\left( z \mapsto \frac{e^{- \frac{m_{0}^{2}}{2 s_{0}^{2}}} e^{- \frac{m_{1}^{2}}{2 s_{1}^{2}}} \int\limits_{-\infty}^{\infty} e^{- \frac{x_{1}^{2}}{2 s_{1}^{2}}} e^{\frac{m_{1} x_{1}}{s_{1}^{2}}} \int\limits_{-\infty}^{\infty} e^{- \frac{x_{0}^{2}}{2 s_{0}^{2}}} e^{\frac{m_{0} x_{0}}{s_{0}^{2}}} \delta\left(- z + \frac{m_{0} m_{1}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right|} - \frac{m_{0} x_{1}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right|} - \frac{m_{1} x_{0}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right|} + \frac{x_{0} x_{1}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right|}\right)\, dx_{0}\, dx_{1}}{2 \pi s_{0} s_{1}} \right)\]
\[\left( z \mapsto \frac{e^{- \frac{m_{0}^{2}}{2 s_{0}^{2}}} e^{- \frac{m_{1}^{2}}{2 s_{1}^{2}}} \int\limits_{-\infty}^{\infty} e^{- \frac{x_{1}^{2}}{2 s_{1}^{2}}} e^{\frac{m_{1} x_{1}}{s_{1}^{2}}} \int\limits_{-\infty}^{\infty} e^{- \frac{x_{0}^{2}}{2 s_{0}^{2}}} e^{\frac{m_{0} x_{0}}{s_{0}^{2}}} \delta\left(- z + \frac{x_{0} x_{1}}{\left|{x_{0}}\right| \left|{x_{1}}\right|}\right)\, dx_{0}\, dx_{1}}{2 \pi s_{0} s_{1}} \right)\]
\[\left( z \mapsto \frac{e^{- \frac{m_{0}^{2}}{2 s_{0}^{2}}} e^{- \frac{m_{1}^{2}}{2 s_{1}^{2}}} \int\limits_{-\infty}^{\infty} e^{- \frac{x_{1}^{2}}{2 s_{1}^{2}}} e^{\frac{m_{1} x_{1}}{s_{1}^{2}}} \int\limits_{-\infty}^{\infty} e^{- \frac{x_{0}^{2}}{2 s_{0}^{2}}} e^{\frac{m_{0} x_{0}}{s_{0}^{2}}} \delta\left(- z + \frac{m_{0} m_{1}}{\left|{m_{0} - x_{0}}\right|^{1.0} \left|{m_{1} - x_{1}}\right|^{1.0}} - \frac{m_{0} x_{1}}{\left|{m_{0} - x_{0}}\right|^{1.0} \left|{m_{1} - x_{1}}\right|^{1.0}} - \frac{m_{1} x_{0}}{\left|{m_{0} - x_{0}}\right|^{1.0} \left|{m_{1} - x_{1}}\right|^{1.0}} + \frac{x_{0} x_{1}}{\left|{m_{0} - x_{0}}\right|^{1.0} \left|{m_{1} - x_{1}}\right|^{1.0}}\right)\, dx_{0}\, dx_{1}}{2 \pi s_{0} s_{1}} \right)\]
I have no idea if these numbers are AM or PM, but I guess I calculated these expressions for increasing numbers of variables and recorded when they completed:
| NVars | Time |
|---|---|
| 2 | 08:50:47 |
| 3 | 08:50:48 |
| 4 | 08:50:49 |
| 5 | 08:50:53 |
| 6 | 08:51:01 |
| 7 | 08:51:14 |
| 8 | 08:51:35 |
| 9 | 08:52:06 |
| 10 | 08:52:52 |
| 11 | 08:53:57 |
| 12 | 08:55:25 |
| 13 | 08:57:24 |
| 14 | 09:00:02 |
| 15 | 09:03:34 |
And here is some code I found which I guess tries to put together ratio distributions involving normal variables.
"""Core helpers for the insane PDFs post."""
from __future__ import annotations
def generate() -> str:
"""Return a short marker string matching the script intent."""
return "INSANE_PDFS"I think the following might be the generated output:
Sun Dec 27 17:44:45 2020 [x0, x1] \[\left( z \mapsto \frac{e^{- \frac{m_{0}^{2}}{2 s_{0}^{2}}} e^{- \frac{m_{1}^{2}}{2 s_{1}^{2}}} \int\limits_{-\infty}^{\infty} e^{- \frac{x_{1}^{2}}{2 s_{1}^{2}}} e^{\frac{m_{1} x_{1}}{s_{1}^{2}}} \int\limits_{-\infty}^{\infty} e^{- \frac{x_{0}^{2}}{2 s_{0}^{2}}} e^{\frac{m_{0} x_{0}}{s_{0}^{2}}} \delta\left(- z + \frac{m_{0} m_{1}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right|} - \frac{m_{0} x_{1}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right|} - \frac{m_{1} x_{0}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right|} + \frac{x_{0} x_{1}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right|}\right)\, dx_{0}\, dx_{1}}{2 \pi s_{0} s_{1}} \right)\]
Sun Dec 27 17:44:47 2020 [x0, x1, x2] \[\left( z \mapsto \frac{\sqrt{2} e^{- \frac{m_{0}^{2}}{2 s_{0}^{2}}} e^{- \frac{m_{1}^{2}}{2 s_{1}^{2}}} e^{- \frac{m_{2}^{2}}{2 s_{2}^{2}}} \int\limits_{-\infty}^{\infty} e^{- \frac{x_{2}^{2}}{2 s_{2}^{2}}} e^{\frac{m_{2} x_{2}}{s_{2}^{2}}} \int\limits_{-\infty}^{\infty} e^{- \frac{x_{1}^{2}}{2 s_{1}^{2}}} e^{\frac{m_{1} x_{1}}{s_{1}^{2}}} \int\limits_{-\infty}^{\infty} e^{- \frac{x_{0}^{2}}{2 s_{0}^{2}}} e^{\frac{m_{0} x_{0}}{s_{0}^{2}}} \delta\left(- z - \frac{m_{0} m_{1} m_{2}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right|} + \frac{m_{0} m_{1} x_{2}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right|} + \frac{m_{0} m_{2} x_{1}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right|} - \frac{m_{0} x_{1} x_{2}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right|} + \frac{m_{1} m_{2} x_{0}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right|} - \frac{m_{1} x_{0} x_{2}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right|} - \frac{m_{2} x_{0} x_{1}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right|} + \frac{x_{0} x_{1} x_{2}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right|}\right)\, dx_{0}\, dx_{1}\, dx_{2}}{4 \pi^{\frac{3}{2}} s_{0} s_{1} s_{2}} \right)\]
Sun Dec 27 17:44:52 2020 [x0, x1, x2, x3] \[\left( z \mapsto \frac{e^{- \frac{m_{0}^{2}}{2 s_{0}^{2}}} e^{- \frac{m_{1}^{2}}{2 s_{1}^{2}}} e^{- \frac{m_{2}^{2}}{2 s_{2}^{2}}} e^{- \frac{m_{3}^{2}}{2 s_{3}^{2}}} \int\limits_{-\infty}^{\infty} e^{- \frac{x_{3}^{2}}{2 s_{3}^{2}}} e^{\frac{m_{3} x_{3}}{s_{3}^{2}}} \int\limits_{-\infty}^{\infty} e^{- \frac{x_{2}^{2}}{2 s_{2}^{2}}} e^{\frac{m_{2} x_{2}}{s_{2}^{2}}} \int\limits_{-\infty}^{\infty} e^{- \frac{x_{1}^{2}}{2 s_{1}^{2}}} e^{\frac{m_{1} x_{1}}{s_{1}^{2}}} \int\limits_{-\infty}^{\infty} e^{- \frac{x_{0}^{2}}{2 s_{0}^{2}}} e^{\frac{m_{0} x_{0}}{s_{0}^{2}}} \delta\left(- z + \frac{m_{0} m_{1} m_{2} m_{3}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right|} - \frac{m_{0} m_{1} m_{2} x_{3}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right|} - \frac{m_{0} m_{1} m_{3} x_{2}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right|} + \frac{m_{0} m_{1} x_{2} x_{3}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right|} - \frac{m_{0} m_{2} m_{3} x_{1}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right|} + \frac{m_{0} m_{2} x_{1} x_{3}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right|} + \frac{m_{0} m_{3} x_{1} x_{2}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right|} - \frac{m_{0} x_{1} x_{2} x_{3}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right|} - \frac{m_{1} m_{2} m_{3} x_{0}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right|} + \frac{m_{1} m_{2} x_{0} x_{3}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right|} + \frac{m_{1} m_{3} x_{0} x_{2}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right|} - \frac{m_{1} x_{0} x_{2} x_{3}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right|} + \frac{m_{2} m_{3} x_{0} x_{1}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right|} - \frac{m_{2} x_{0} x_{1} x_{3}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right|} - \frac{m_{3} x_{0} x_{1} x_{2}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right|} + \frac{x_{0} x_{1} x_{2} x_{3}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right|}\right)\, dx_{0}\, dx_{1}\, dx_{2}\, dx_{3}}{4 \pi^{2} s_{0} s_{1} s_{2} s_{3}} \right)\]
Sun Dec 27 17:45:12 2020 [x0, x1, x2, x3, x4] \[\left( z \mapsto \frac{\sqrt{2} e^{- \frac{m_{0}^{2}}{2 s_{0}^{2}}} e^{- \frac{m_{1}^{2}}{2 s_{1}^{2}}} e^{- \frac{m_{2}^{2}}{2 s_{2}^{2}}} e^{- \frac{m_{3}^{2}}{2 s_{3}^{2}}} e^{- \frac{m_{4}^{2}}{2 s_{4}^{2}}} \int\limits_{-\infty}^{\infty} e^{- \frac{x_{4}^{2}}{2 s_{4}^{2}}} e^{\frac{m_{4} x_{4}}{s_{4}^{2}}} \int\limits_{-\infty}^{\infty} e^{- \frac{x_{3}^{2}}{2 s_{3}^{2}}} e^{\frac{m_{3} x_{3}}{s_{3}^{2}}} \int\limits_{-\infty}^{\infty} e^{- \frac{x_{2}^{2}}{2 s_{2}^{2}}} e^{\frac{m_{2} x_{2}}{s_{2}^{2}}} \int\limits_{-\infty}^{\infty} e^{- \frac{x_{1}^{2}}{2 s_{1}^{2}}} e^{\frac{m_{1} x_{1}}{s_{1}^{2}}} \int\limits_{-\infty}^{\infty} e^{- \frac{x_{0}^{2}}{2 s_{0}^{2}}} e^{\frac{m_{0} x_{0}}{s_{0}^{2}}} \delta\left(- z - \frac{m_{0} m_{1} m_{2} m_{3} m_{4}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right|} + \frac{m_{0} m_{1} m_{2} m_{3} x_{4}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right|} + \frac{m_{0} m_{1} m_{2} m_{4} x_{3}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right|} - \frac{m_{0} m_{1} m_{2} x_{3} x_{4}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right|} + \frac{m_{0} m_{1} m_{3} m_{4} x_{2}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right|} - \frac{m_{0} m_{1} m_{3} x_{2} x_{4}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right|} - \frac{m_{0} m_{1} m_{4} x_{2} x_{3}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right|} + \frac{m_{0} m_{1} x_{2} x_{3} x_{4}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right|} + \frac{m_{0} m_{2} m_{3} m_{4} x_{1}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right|} - \frac{m_{0} m_{2} m_{3} x_{1} x_{4}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right|} - \frac{m_{0} m_{2} m_{4} x_{1} x_{3}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right|} + \frac{m_{0} m_{2} x_{1} x_{3} x_{4}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right|} - \frac{m_{0} m_{3} m_{4} x_{1} x_{2}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right|} + \frac{m_{0} m_{3} x_{1} x_{2} x_{4}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right|} + \frac{m_{0} m_{4} x_{1} x_{2} x_{3}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right|} - \frac{m_{0} x_{1} x_{2} x_{3} x_{4}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right|} + \frac{m_{1} m_{2} m_{3} m_{4} x_{0}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right|} - \frac{m_{1} m_{2} m_{3} x_{0} x_{4}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right|} - \frac{m_{1} m_{2} m_{4} x_{0} x_{3}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right|} + \frac{m_{1} m_{2} x_{0} x_{3} x_{4}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right|} - \frac{m_{1} m_{3} m_{4} x_{0} x_{2}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right|} + \frac{m_{1} m_{3} x_{0} x_{2} x_{4}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right|} + \frac{m_{1} m_{4} x_{0} x_{2} x_{3}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right|} - \frac{m_{1} x_{0} x_{2} x_{3} x_{4}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right|} - \frac{m_{2} m_{3} m_{4} x_{0} x_{1}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right|} + \frac{m_{2} m_{3} x_{0} x_{1} x_{4}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right|} + \frac{m_{2} m_{4} x_{0} x_{1} x_{3}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right|} - \frac{m_{2} x_{0} x_{1} x_{3} x_{4}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right|} + \frac{m_{3} m_{4} x_{0} x_{1} x_{2}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right|} - \frac{m_{3} x_{0} x_{1} x_{2} x_{4}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right|} - \frac{m_{4} x_{0} x_{1} x_{2} x_{3}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right|} + \frac{x_{0} x_{1} x_{2} x_{3} x_{4}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right|}\right)\, dx_{0}\, dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}}{8 \pi^{\frac{5}{2}} s_{0} s_{1} s_{2} s_{3} s_{4}} \right)\]
Sun Dec 27 17:46:48 2020 [x0, x1, x2, x3, x4, x5] \[\left( z \mapsto \frac{e^{- \frac{m_{0}^{2}}{2 s_{0}^{2}}} e^{- \frac{m_{1}^{2}}{2 s_{1}^{2}}} e^{- \frac{m_{2}^{2}}{2 s_{2}^{2}}} e^{- \frac{m_{3}^{2}}{2 s_{3}^{2}}} e^{- \frac{m_{4}^{2}}{2 s_{4}^{2}}} e^{- \frac{m_{5}^{2}}{2 s_{5}^{2}}} \int\limits_{-\infty}^{\infty} e^{- \frac{x_{5}^{2}}{2 s_{5}^{2}}} e^{\frac{m_{5} x_{5}}{s_{5}^{2}}} \int\limits_{-\infty}^{\infty} e^{- \frac{x_{4}^{2}}{2 s_{4}^{2}}} e^{\frac{m_{4} x_{4}}{s_{4}^{2}}} \int\limits_{-\infty}^{\infty} e^{- \frac{x_{3}^{2}}{2 s_{3}^{2}}} e^{\frac{m_{3} x_{3}}{s_{3}^{2}}} \int\limits_{-\infty}^{\infty} e^{- \frac{x_{2}^{2}}{2 s_{2}^{2}}} e^{\frac{m_{2} x_{2}}{s_{2}^{2}}} \int\limits_{-\infty}^{\infty} e^{- \frac{x_{1}^{2}}{2 s_{1}^{2}}} e^{\frac{m_{1} x_{1}}{s_{1}^{2}}} \int\limits_{-\infty}^{\infty} e^{- \frac{x_{0}^{2}}{2 s_{0}^{2}}} e^{\frac{m_{0} x_{0}}{s_{0}^{2}}} \delta\left(- z + \frac{m_{0} m_{1} m_{2} m_{3} m_{4} m_{5}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} - \frac{m_{0} m_{1} m_{2} m_{3} m_{4} x_{5}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} - \frac{m_{0} m_{1} m_{2} m_{3} m_{5} x_{4}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} + \frac{m_{0} m_{1} m_{2} m_{3} x_{4} x_{5}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} - \frac{m_{0} m_{1} m_{2} m_{4} m_{5} x_{3}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} + \frac{m_{0} m_{1} m_{2} m_{4} x_{3} x_{5}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} + \frac{m_{0} m_{1} m_{2} m_{5} x_{3} x_{4}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} - \frac{m_{0} m_{1} m_{2} x_{3} x_{4} x_{5}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} - \frac{m_{0} m_{1} m_{3} m_{4} m_{5} x_{2}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} + \frac{m_{0} m_{1} m_{3} m_{4} x_{2} x_{5}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} + \frac{m_{0} m_{1} m_{3} m_{5} x_{2} x_{4}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} - \frac{m_{0} m_{1} m_{3} x_{2} x_{4} x_{5}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} + \frac{m_{0} m_{1} m_{4} m_{5} x_{2} x_{3}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} - \frac{m_{0} m_{1} m_{4} x_{2} x_{3} x_{5}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} - \frac{m_{0} m_{1} m_{5} x_{2} x_{3} x_{4}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} + \frac{m_{0} m_{1} x_{2} x_{3} x_{4} x_{5}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} - \frac{m_{0} m_{2} m_{3} m_{4} m_{5} x_{1}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} + \frac{m_{0} m_{2} m_{3} m_{4} x_{1} x_{5}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} + \frac{m_{0} m_{2} m_{3} m_{5} x_{1} x_{4}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} - \frac{m_{0} m_{2} m_{3} x_{1} x_{4} x_{5}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} + \frac{m_{0} m_{2} m_{4} m_{5} x_{1} x_{3}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} - \frac{m_{0} m_{2} m_{4} x_{1} x_{3} x_{5}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} - \frac{m_{0} m_{2} m_{5} x_{1} x_{3} x_{4}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} + \frac{m_{0} m_{2} x_{1} x_{3} x_{4} x_{5}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} + \frac{m_{0} m_{3} m_{4} m_{5} x_{1} x_{2}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} - \frac{m_{0} m_{3} m_{4} x_{1} x_{2} x_{5}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} - \frac{m_{0} m_{3} m_{5} x_{1} x_{2} x_{4}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} + \frac{m_{0} m_{3} x_{1} x_{2} x_{4} x_{5}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} - \frac{m_{0} m_{4} m_{5} x_{1} x_{2} x_{3}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} + \frac{m_{0} m_{4} x_{1} x_{2} x_{3} x_{5}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} + \frac{m_{0} m_{5} x_{1} x_{2} x_{3} x_{4}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} - \frac{m_{0} x_{1} x_{2} x_{3} x_{4} x_{5}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} - \frac{m_{1} m_{2} m_{3} m_{4} m_{5} x_{0}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} + \frac{m_{1} m_{2} m_{3} m_{4} x_{0} x_{5}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} + \frac{m_{1} m_{2} m_{3} m_{5} x_{0} x_{4}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} - \frac{m_{1} m_{2} m_{3} x_{0} x_{4} x_{5}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} + \frac{m_{1} m_{2} m_{4} m_{5} x_{0} x_{3}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} - \frac{m_{1} m_{2} m_{4} x_{0} x_{3} x_{5}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} - \frac{m_{1} m_{2} m_{5} x_{0} x_{3} x_{4}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} + \frac{m_{1} m_{2} x_{0} x_{3} x_{4} x_{5}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} + \frac{m_{1} m_{3} m_{4} m_{5} x_{0} x_{2}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} - \frac{m_{1} m_{3} m_{4} x_{0} x_{2} x_{5}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} - \frac{m_{1} m_{3} m_{5} x_{0} x_{2} x_{4}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} + \frac{m_{1} m_{3} x_{0} x_{2} x_{4} x_{5}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} - \frac{m_{1} m_{4} m_{5} x_{0} x_{2} x_{3}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} + \frac{m_{1} m_{4} x_{0} x_{2} x_{3} x_{5}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} + \frac{m_{1} m_{5} x_{0} x_{2} x_{3} x_{4}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} - \frac{m_{1} x_{0} x_{2} x_{3} x_{4} x_{5}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} + \frac{m_{2} m_{3} m_{4} m_{5} x_{0} x_{1}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} - \frac{m_{2} m_{3} m_{4} x_{0} x_{1} x_{5}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} - \frac{m_{2} m_{3} m_{5} x_{0} x_{1} x_{4}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} + \frac{m_{2} m_{3} x_{0} x_{1} x_{4} x_{5}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} - \frac{m_{2} m_{4} m_{5} x_{0} x_{1} x_{3}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} + \frac{m_{2} m_{4} x_{0} x_{1} x_{3} x_{5}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} + \frac{m_{2} m_{5} x_{0} x_{1} x_{3} x_{4}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} - \frac{m_{2} x_{0} x_{1} x_{3} x_{4} x_{5}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} - \frac{m_{3} m_{4} m_{5} x_{0} x_{1} x_{2}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} + \frac{m_{3} m_{4} x_{0} x_{1} x_{2} x_{5}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} + \frac{m_{3} m_{5} x_{0} x_{1} x_{2} x_{4}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} - \frac{m_{3} x_{0} x_{1} x_{2} x_{4} x_{5}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} + \frac{m_{4} m_{5} x_{0} x_{1} x_{2} x_{3}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} - \frac{m_{4} x_{0} x_{1} x_{2} x_{3} x_{5}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} - \frac{m_{5} x_{0} x_{1} x_{2} x_{3} x_{4}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|} + \frac{x_{0} x_{1} x_{2} x_{3} x_{4} x_{5}}{\left|{m_{0} - x_{0}}\right| \left|{m_{1} - x_{1}}\right| \left|{m_{2} - x_{2}}\right| \left|{m_{3} - x_{3}}\right| \left|{m_{4} - x_{4}}\right| \left|{m_{5} - x_{5}}\right|}\right)\, dx_{0}\, dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}\, dx_{5}}{8 \pi^{3} s_{0} s_{1} s_{2} s_{3} s_{4} s_{5}} \right)\]
I have a feeling like the outputs of this code didn’t end up answering my research questions, but they do remind me that academic work is full of failures. That’s okay. It is really difficult to figure things out and iteratively trying different approaches helps us learn (and sometimes succeed).