Integrating Feynman integrals in mathematica
- To: mathgroup at smc.vnet.net
- Subject: [mg50839] Integrating Feynman integrals in mathematica
- From: pabird at supanet.com (Xman)
- Date: Thu, 23 Sep 2004 05:27:21 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
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).