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\begin{document}

\section{Feynman Parameters}
\beqa
{{1}\over{A_1^{m_1}...A_n^{m_n}}} &=&
\int_0^\infinity dx_1...dx_n\, \delta(x_1+...x_n-1)
{{x_1^{m_1-1}...x_n^{m_n-1}}\over{\left(x_1 A_1+...+x_n A_n
\right)^{m_1+...+m_n}}}
{{\Gamma(m_1+...+m_n)}\over{\Gamma(m_1)...\Gamma(m_n)}}
\nonumber
\eeqa

\beqa
{{1}\over{A^n}} - {{1}\over{B^n}}&=&
\int_0^\infinity dx dy\, \delta(x+y-1) 
{{n(B-A)}\over{\left(x A+y B \right)^{n+1}}}
\nonumber
\eeqa

\section{Dim. Reg.}

\beqa
\int {{d^{2\omega}k}\over{(2\pi)^{2\omega}}} \,
{1\over{\left(k^2-\Delta\right)^r}}&=&
i\, (-1)^r {{\Gamma(r-\omega)}\over{(4 \pi)^\omega \,\Gamma(r)}}
\,\Delta^{\omega-r}
\nonumber
\eeqa

\beqa
\int {{d^{2\omega}k}\over{(2\pi)^{2\omega}}} \,
{{k^\mu k^\nu}\over{\left(k^2-\Delta\right)^r}}&=&
i\, (-1)^{1+r} \,{{\Gamma(r-\omega-1)}\over{2\,(4 \pi)^\omega \,\Gamma(r)}}
\,\Delta^{\omega+1-r}\,g^{\mu \nu}
\nonumber
\eeqa

\section{Shifting}

\beqa
\int{{d^4p}\over{(2\pi)^4}} {{p^\mu}\over{((p+q)^2 -m^2)^2}}
&=& \int{{d^4p}\over{(2\pi)^4}} {{(p-q)^\mu}\over{(p^2 -m^2)^2}}
+{{i\, q^\mu}\over{32 \pi^2}} 
\nonumber
\eeqa

\beqa
\int{{d^4p}\over{(2\pi)^4}} 
{{p^\mu p^\nu p^\tau}\over{((p+q)^2 -m^2)^3}}
&=& \int{{d^4p}\over{(2\pi)^4}} 
{{(p-q)^\mu (p-q)^\nu (p-q)^\tau}\over{(p^2 -m^2)^3}}\nonumber\\
& &+{{i }\over{192 \pi^2}}
\left(g^{\mu\nu}q^\tau +g^{\nu\tau}q^\mu +g^{\mu\tau}q^\nu \right)
\nonumber
\eeqa
\beqa
\int{{d^4p}\over{(2\pi)^4}} 
{{p^\mu p^\nu}\over{((p+q)^2 -m^2)^2}}
&=& \int{{d^4p}\over{(2\pi)^4}} 
{{(p-q)^\mu (p-q)^\nu}\over{(p^2 -m^2)^2}}\nonumber\\
& &-{{i \,q^\mu q^\nu}\over{32 \pi^2}}
-{{i }\over{96 \pi^2}}
\left(g^{\mu\nu}q^2 + 2 q^\mu q^\nu \right)
\nonumber
\eeqa

\section{Miscellaneous}

\beqa
{1\over{4 \pi}} \int d\Omega_k {\cal J}(\bf{v},\bf{v^\prime}) &=&
\int_0^1 dx\,{{q^2\left(2x(1-x)-1\right)}\over{m^2-q^2 x(1-x)}}
\nonumber
\eeqa

where
\beqa
{\cal J}(\bf{v},\bf{v^\prime}) &\equiv&
\frac{2p.p^\prime}
{ (E- {\bf p} .{\bf \hat k}  )   ( E^\prime-{\bf p^\prime} .{\bf \hat k} )  }
-\frac{m^2}
{(E-{\bf p} .{\bf \hat k} )  (E-{\bf p} .{\bf \hat k} )}
-\frac{m^2}
{(E^\prime-{\bf p^\prime} .{\bf \hat k} )(E^\prime-{\bf p^\prime} .{\bf \hat k} )}
\nonumber
\eeqa
and $p^\prime = p + q$, $p=E(1,\bf{v})$, $p^\prime=E^\prime(1,\bf{v^\prime})$

\end{document}





