Re: Integrating Feynman integrals in mathematica

*To*: mathgroup at smc.vnet.net*Subject*: [mg50854] Re: Integrating Feynman integrals in mathematica*From*: "Steve Luttrell" <steve_usenet at _removemefirst_luttrell.org.uk>*Date*: Fri, 24 Sep 2004 04:41:27 -0400 (EDT)*References*: <ciu5iv$81d$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Unless I have missed the point of your question, this is standard material covered in textbooks on quantum field theory. I have Itzykson + Zuber's tome, and I find your first integral discussed in section 1-3-1 on Green functions. Your second integral is related to your first integral by the trick of differentiating w.r.t. an auxiliary parameter (i.e. make the replacement m^2-->a m^2 in your first integral, differentiate w.r.t. a, then set a-->1). Steve Luttrell "Xman" <pabird at supanet.com> wrote in message news:ciu5iv$81d$1 at smc.vnet.net... >I am trying to find an explicit form of the following 4-dimensional > fourier transforms. Can anyone help? ( x and k are 4 dimensional > vectors) They are from physics. > > 1) > > f(x) =Intregral[ e^(i x.k) / (k.k -m^2) ]dk^4 > > 2) > > g(x)=Intregral[ e^(i x.k) / (k.k -m^2)^2 ]dk^4 > > I know that the first is of the form: > > f(x) = 1/|x.x| + log|x.x| * P((m^2/4) |x.x|) + Q((m^2/4) |x.x|) > > (when m=0 this becomes 1/|x.x|) > > Where P and Q stand for infinite polynomial series and that I think > P(y) = Sum( y^n /(n!(n+1)!) ,y=0..infinity ) > > and that in the second one > g(x) = log|x.x| * R((m^2/4) |x.x|) + S((m^2/4) |x.x|) > > (when m=0 this becomes log|x.x|) > > where R(y) = Sum( y^n /(n!n!) ,y=0..infinity ) > > But the functions Q and S are more difficult to find. > Plus does anyone know if the series P and R (=P') or Q and S can be > written in terms of simple functions? > > It may help to know that f and g satisfy the following 4 dimensional > wave equations: > > ( d/dx . d/dx - m^2) f(x) = delta(x) (=0 for x=/=0) > ( d/dx . d/dx - m^2)^2 g(x) = delta(x) (=0 for x=/=0) > > I am particularly interested in g(x). >