1+1

DG=(-w^2 +I lbd w + k);
G=1/DG

Ree[z_]:=
Coefficient[Factor[ExpandAll[z]/.{Complex[0,r_]->r Iz}],Iz,0];
Ima[z_]:=
Coefficient[Factor[ExpandAll[z]/.{Complex[0,r_]->r Iz}],Iz,1];
Conj[z_]:=Ree[z]-I Ima[z];

Ree[DG]

dG=D[G,w]

Ima[dG]

Solve[Im[]==0,w]

G

Plot[Im[G/.{lbd->.5,k->1}],{w,0,2}]