$Assumptions={σ>0, x0 ∈Reals, n0∈Reals}

ξ[x_,x0_,σ_]=1/Sqrt[2Pi σ^2]Exp[-(x-x0)^2/(2 σ^2)]

(** With negative frequencies **)

norm = Integrate[ξ[x,x0,σ],{x,-Infinity,+Infinity}]

gT = Integrate[ξ[x,0.0,σ] /norm Exp[-I x T] ,{x,-Infinity,+Infinity},Assumptions->T∈Reals]//FullSimplify

(** Now excluding negative frequencies **)

norm = Integrate[ξ[x,x0,σ],{x,0,+Infinity}]//FullSimplify

gτ = Integrate[ξ[x,x0,σ]/norm Exp[-I x τ] ,{x,0,+Infinity},Assumptions->τ∈Reals]//FullSimplify

(** Now using a differentiable function **)

norm = Integrate[x * ξ[x,x0,σ],{x,0,+Infinity}]//FullSimplify

gτ2 = Integrate[x * ξ[x,x0,σ]/norm Exp[-I x τ] ,{x,0,+Infinity},Assumptions->τ∈Reals]//FullSimplify

Plot[Evaluate[{gτ,gτ2}/.{x0->0.3, σ->0.3}//Re],{τ,-30,30},PlotRange->Full]