User talk:Xetrov

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Greetings, Jeff3000. Thank you kindly for the resource-laden message. Xetrov 23:10, 21 February 2007 (UTC)

Gaussian_integral#Higher-order_polynomials

http://en.wikipedia.org/wiki/Gaussian_integral#Higher-order_polynomials

Hello,

Do you have any references for the formula given (or the general technique of using series approximations of Gaussian integrals)? Would you happen to know what aspect of quantum field theory involves these integrals? I searched a lot for these things, but the terms Gaussian and integral are so general that I'm not getting anywhere without more specific information.

Thanks in advance! --Xeṭrov 01:24, 24 January 2011 (UTC)

I don't know any references for the formula, but it follows from the definition of Γ, using $t\equiv ax^{n},\quad {\rm {d}}t={\rm {d}}x\,anx^{n-1},\quad z\equiv {\frac {j+1}{n}}$:

$\Gamma (z)\equiv \int _{0}^{\infty }{\rm {d}}t\,t^{z-1}e^{-t}$

$\Gamma \left({\frac {j+1}{n}}\right)=\int _{0}^{\infty }{\rm {d}}x\,a^{\frac {j+1}{n}}nx^{j}e^{-ax^{n}}$

$\int _{0}^{\infty }{\rm {d}}x\,x^{j}e^{-ax^{n}}={\frac {1}{n}}{\frac {\Gamma \left({\frac {j+1}{n}}\right)}{a^{\frac {j+1}{n}}}}$

and expanding $\int _{0}^{\infty }{\rm {d}}x\,e^{-ax^{n}+bx^{n-1}+\dots +cx}$ in terms of $b\dots c$ gives

$\int _{0}^{\infty }{\rm {d}}x\,e^{-ax^{n}+bx^{n-1}+\dots +cx}=\sum _{B\dots C=0}^{\infty }{\frac {b^{B}}{B!}}\dots {\frac {c^{C}}{C!}}\int _{0}^{\infty }{\rm {d}}x\,x^{(n-1)B+\dots +C}e^{-ax^{n}}$

$\int _{0}^{\infty }{\rm {d}}x\,e^{-ax^{n}+bx^{n-1}+\dots +cx}={\frac {1}{n}}\sum _{B\dots C=0}^{\infty }{\frac {b^{B}}{B!}}\dots {\frac {c^{C}}{C!}}{\frac {\Gamma \left({\frac {(n-1)B+\dots +C+1}{n}}\right)}{a^{\frac {(n-1)B+\dots +C+1}{n}}}}$.

Path integrals are usually gaussian integrals or similar. Κσυπ Cyp   18:28, 25 January 2011 (UTC)

Thank you very much for explaining this. It looks like approximations are in order, but it's good to know that there is an exact formula. --Xeṭrov 22:47, 27 January 2011 (UTC)

What happens to $x^{j}$ after the "and expanding [...] in terms of" line? I assume the sums come from the infinite sum expansions of $e^{[term]}$. Is it possible to expand $e^{-ax^{n}}$ so each term is just a product of expansion terms? --Xetrov 01:54, 2 February 2011 (UTC)

Oh, sorry, that $x^{j}$ wasn't supposed to be there (if it was there, the j would be added to the ${\frac {(n-1)B+\dots +C+1}{n}}+j$ terms), as in:

$\int _{0}^{\infty }{\rm {d}}x\,x^{j}e^{-ax^{n}+bx^{n-1}+\dots +cx}={\frac {1}{n}}\sum _{B\dots C=0}^{\infty }{\frac {b^{B}}{B!}}\dots {\frac {c^{C}}{C!}}{\frac {\Gamma \left({\frac {(n-1)B+\dots +C+1}{n}}+j\right)}{a^{{\frac {(n-1)B+\dots +C+1}{n}}+j}}}$

If also expanding the $e^{-ax^{n}}$ term, then all terms would be of the form $\int _{0}^{\infty }{\rm {d}}x\,x^{A}=\infty$. If expanding the $e^{-ax^{n}}$ term but leaving the $e^{-bx^{m}}$ term (with $m<n$, then the sum would not converge, due to $\Gamma \left({\frac {nA+\dots }{m}}\right)$ growing faster than $A!$. Κσυπ Cyp   02:04, 3 February 2011 (UTC)