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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: ownerwrimathgroup at wolfram.com
I am trying to find an explicit form of the following 4dimensional
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 + logx.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) = logx.x * R((m^2/4) x.x) + S((m^2/4) x.x)
(when m=0 this becomes logx.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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