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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])

If I add assumptions,

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.

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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]) *)
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