<<Statistics`MultinormalDistribution`

\(\(Sqrt[
          PDF[MultinormalDistribution[{rx, 
                ry}, {{ϵ^2, 0}, {0, ϵ^2}}], {x, y}]]
 // 
        PowerExpand\) // Simplify\)

StandardDeviation[
      MultinormalDistribution[{rx, 
          ry}, {{ϵ^2, 0}, {0, ϵ^2}}]]

<< Calculus`FourierTransform`

ψ1[x_] = 
      Exp[-(x - (Subscript[r, x]))^2/(4  ϵ^2)]/
          (Sqrt[2  Pi] ϵ) Exp[I x (Subscript[k, x])]

FourierTransform[ψ1[x], x, q] // ComplexExpand

Integrate[
      ψ1[x] Exp[2  Pi I f x] // PowerExpand, {x, -Infinity, 
        Infinity}]

FourierTransform[Exp[-(x - rx)^2/(4  ϵ^2)]]