# Convolve doesn't yeild expected answer, nor does integrate

I am convolving, or trying to, a Gaussian and a Lorentzian as follows:

Gaussian1[ \[Mu]_, \[Sigma]_, x_] :=
1/(\[Sigma] Sqrt[2 \[Pi]]) Exp[-(1/2) ((x - \[Mu]) / \[Sigma])^2]

Lorentzian[\[Gamma]_, x0_, x_] :=
1/\[Pi] (1/2 \[Gamma])/((x - x0)^2 + (1/2 \[Gamma])^2)


The convolution attempt being

Convolve[Lorentzian[1, \[Gamma], x0, x], Gaussian1[1, \[Mu], \[Sigma], x], x, y]


However this simply returns itself except output formatted:

 (\[Gamma] Convolve[1/((x - x0)^2 + \[Gamma]^2/4),
E^(-((x - \[Mu])^2/(2 \[Sigma]^2))), x, y])/(2 Sqrt[2] \[Pi]^(
3/2) \[Sigma])


Assumptions->Re[\[Gamma]] && Re[\[Sigma]]


The result is the same. I also tried to perform the convolution using Integrate[] as:

Integrate[
1/\[Pi] (1/2 \[Gamma])/((\[Tau])^2 + (1/2 \[Gamma])^2) * 1/(\[Sigma] Sqrt[2 \[Pi]]) Exp[-(1/2) ((t - \[Tau]) / \[Sigma])^2],
{\[Tau], -Infinity, +Infinity},
Assumptions->Re[\[Gamma]] && Re[\[Sigma]]
]


No joy! This convolution is well known as a Voigt profile So it has a solution but I don't understand what I am doing wrong.

You could try recasting your formulae to the internal PDF distributions NormalDistribution and CauchyDistribution.

Convolve[PDF[NormalDistribution[0, σ], x],
PDF[CauchyDistribution[a, b], x], x, y]
(* -(1/(2 Sqrt[2] π^(3/2) σ))
I E^((-a^2 + (b - I y)^2 + 2 a (-I b + y))/(
2 σ^2)) (E^((
2 I a b)/σ^2) π Erfi[((a - I b - y) Sqrt[
1/σ^2])/Sqrt[2]] -
E^((2 I b y)/σ^2) π Erfi[((a + I b - y) Sqrt[
1/σ^2])/Sqrt[2]] -
E^((2 I a b)/σ^2) Log[a - I b - y] +
E^((2 I b y)/σ^2) Log[a + I b - y] -
E^((2 I b y)/σ^2) Log[-a - I b + y] +
E^((2 I a b)/σ^2) Log[-a + I b + y]) *)