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MathGroup Archive 2004

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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).


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