Re: Mathematica can't calculate Fourier transform (Dirac mean position eigenfunction)

*To*: mathgroup at smc.vnet.net*Subject*: [mg54988] Re: Mathematica can't calculate Fourier transform (Dirac mean position eigenfunction)*From*: dh <dh at metrohm.ch>*Date*: Wed, 9 Mar 2005 06:34:15 -0500 (EST)*References*: <d0juvv$n6i$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Hallo Jacob, in order that a FourierTransform exists, the function must be absolute integrable. Now try to integrate ( (1 + k^2 + (1 + k^2)^(1/2) )^(-1/2) from zero to infinity and you will see that it is diverging. Sincerely, Daniel Jacob wrote: > Hi, I'm attempting to use Mathematica to calculate a mean-position > eigenfunction of the Dirac equation. To do so I need to evaluate > Fourier transforms (from k-space to r-space) of wavefunctions dependent > on: > > ( (1 + k^2 + (1 + k^2)^(1/2) )^(-1/2) > > where k is in units of the Compton wavevector. > > Cell expression: > > Cell[BoxData[ > FractionBox["1", > SqrtBox[ > RowBox[{"1", "+", > SuperscriptBox["k", "2"], "+", > SqrtBox[ > RowBox[{"1", "+", > SuperscriptBox["k", "2"]}]]}]]]], "Output"] > > > Mathematica is unable to evaluate the FT of the above (either Fourier > sine transform or normal FT). Can anyone give any suggestions as to how > I could evaluate it? > > More specifically, I am making a reverse Foldy-Wouthuysen > transformation of a mean-position eigenfunction in p-space, then > transforming the result into r-space assuming spherical symmetry. The > first component of the r-space eigenfunction is given by the Fourier > sine transform of: > > k ( 1 + (1 + k^2)^(-1/2) )^(1/2) > > Cell[BoxData[ > RowBox[{"k", " ", > SqrtBox[ > RowBox[{"1", "+", > FractionBox["1", > SqrtBox[ > RowBox[{"1", "+", > SuperscriptBox["k", "2"]}]]]}]]}]], "Output"] > > > Thanks for any help. >