a = 0.01;
α = 3.1;
I1  =  
        Integrate[((Exp[I*w*x]) - 1)/(x/0.01 + 1)^3.1, {x, 0, 
            Infinity}];
J1  =  -(a/(-1 + α)) + ((a HypergeometricPFQ[{1}, {1 - (α/2), (3/2) - (α/2)}, -(1/4) (a^2) (w^2)])/(-1 + α)) + ((I (a^2) w HypergeometricPFQ[{1}, {(3/2) - (α/2), 2 - (α/2)}, -(1/4) (a^2) (w^2)])/(2 - 3 α + (α^2))) + 
          Gamma[1 - α]*a*
            Exp[-I*w*
                a]*(-I*w*a)^( α - 
                  1);
monster  =  
        I^(1 - α)*(2 - α  -  (1 - α)*
                I);
K1  =  (1 - monster)*a^2*I*
            w/((1 - α)*(2 - α))  +  
          Gamma[1 - α, w*a]*a*
            Exp[-I*w*
                a]*(-I*w*a)^(α - 
                  1);
RI1  =  Re[I1];
II1  =  Im[I1];
RJ1  =  Re[J1];
IJ1  =  Im[J1];
RK1  =  Re[K1];
IK1  =  Im[K1];
Plot[RI1, {w, 0, 1}];
Plot[II1, {w, 0, 1}];
Plot[RJ1, {w, 0, 1}];
Plot[IJ1, {w, 0, 1}];
Plot[RK1, {w, 0, 1}];
Plot[IK1, {w, 0, 1}];
Plot[IJ1 - IK1, {w, 0, 1}];