g[t_] = ((ϵ/π)/((t^2)+(ϵ^2)));
IntegrandV[s_] = ((g[t - s])/(Sqrt[(s^2)-(α^2)]));
IntegrandW[s_] = ((g[t - s])/(Sqrt[(α^2)-(s^2)]));

Z=(Sqrt[(α^2)+(((ϵ-I t))^2)]);
Zconj = (Sqrt[(α^2)+(((ϵ+I t))^2)]);
(* V: the inside of Re() and its conjugate *)
Vin[s_]=(1/(π Z))(-Log[ϵ - I (t-s)] + Log[(α^2)-(t+I ϵ)s - I Z (Sqrt[(s^2)-(α^2)])]);
Vconj[s_]=(1/(π Zconj))(-Log[ϵ + I (t-s)] + Log[(α^2)-(t-I ϵ)s + I Zconj (Sqrt[(s^2)-(α^2)])]);
V[s_]= (1/2)(Vin[s] + Vconj[s]); (* real part *)
(* verify that V is the antiderivative of IntegrandV *)
FullSimplify[D[V[s],s] - IntegrandV[s]]

(* W: the inside of Im() and its conjugate *)
Win[s_]=((-1)/(π Z))(-Log[ϵ - I (t-s)] + Log[(α^2)-(t+I ϵ)s + Z (Sqrt[(α^2)-(s^2)])]);
Wconj[s_]=((-1)/(π Zconj))(-Log[ϵ + I (t-s)] + Log[(α^2)-(t-I ϵ)s + Zconj (Sqrt[(α^2)-(s^2)])]);
W[s_]= (1/(2 I))(Win[s] - Wconj[s]); (* imaginary part *)
(* verify that W is the antiderivative of IntegrandW *)
FullSimplify[D[W[s],s] - IntegrandW[s]]