Gaussian integral

Gauss[1] = FullSimplify[Integrate[Exp[I q x[α] - u q^2 + j[α] q]/( 2 π ), {q, -Infinity, Infinity}, GenerateConditionsu>0], s>0&&u>0&&v>0]

^(j[α] +  x[α])^2/(4 u)/(2 π^(1/2) u^(1/2))

Gauss[2] = (Gauss[1]/.Exp[a_] 1)^d (Gauss[1]/.Exp[a_] Exp[Expand[a]])/(Gauss[1]/.Exp[a_] 1)

2^(-d) ^(j[α]^2/(4 u) + ( j[α] x[α])/(2 u) - x[α]^2/(4 u)) π^(-d/2) u^(-d/2)

Gauss[3] = Gauss[2]/.subsrules[2]//Simplify

2^(-d) ^((1 + ) Abs[j]^2 -  Abs[j - x]^2 - (1 - ) Abs[x]^2)/(4 u) π^(-d/2) u^(-d/2)

Gauss[4] = EvaluatePD[PD[Gauss[3], j[α], j[β]], subsrules[1]]/.{j[α_] 0, j0}//Simplify

2^(-2 - d) ^(-Abs[x]^2/(4 u)) π^(-d/2) u^(-2 - d/2) (2 u K[α, β] - x[α] x[β])

Created by Mathematica  (March 14, 2006)