Xman
Sep23-04, 04:46 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\nI am trying to find an explicit form of the following 4-dimensional\nfourier transforms. Can anyone help? ( x and k are 4 dimensional\nvectors) They are from physics.\n\n1)\n\nf(x) =Intregral[ e^(i x.k) / (k.k -m^2) ]dk^4\n\n2)\n\ng(x)=Intregral[ e^(i x.k) / (k.k -m^2)^2 ]dk^4\n\nI know that the first is of the form:\n\nf(x) = 1/|x.x| + log|x.x| * P((m^2/4) |x.x|) + Q((m^2/4) |x.x|)\n\n(when m=0 this becomes 1/|x.x|)\n\nWhere P and Q stand for infinite polynomial series and that I think\nP(y) = Sum( y^n /(n!(n+1)!) ,y=0..infinity )\n\nand that in the second one\ng(x) = log|x.x| * R((m^2/4) |x.x|) + S((m^2/4) |x.x|)\n\n(when m=0 this becomes log|x.x|)\n\nwhere R(y) = Sum( y^n /(n!n!) ,y=0..infinity )\n\nBut the functions Q and S are more difficult to find.\nPlus does anyone know if the series P and R (=P\') or Q and S can be\nwritten in terms of simple functions?\n\nIt may help to know that f and g satisfy the following 4 dimensional\nwave equations:\n\n( d/dx . d/dx - m^2) f(x) = delta(x) (=0 for x=/=0)\n( d/dx . d/dx - m^2)^2 g(x) = delta(x) (=0 for x=/=0)\n\nI am particularly interested in g(x).\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>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[/itex] for x=/=0)( d/dx . d/dx - m^2)^2 g(x) = \delta(x) (=0 for [itex]x=/=0)
I am particularly interested in g(x).
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[/itex] for x=/=0)( d/dx . d/dx - m^2)^2 g(x) = \delta(x) (=0 for [itex]x=/=0)
I am particularly interested in g(x).