rho[r_,c_,kap_]:=(1-Exp[-c r])^kap
s[r_,k0_,q_]:=Exp[I k0 r] (k0 r)^(-q)
integrand[r_,k_,k0_,c_,kap_,q_,n_]:=rho[r,c,kap] s[r,k0,q] r BesselJ[n,k r]

Lsum30[kap_,c_,k0_,k_]:=Sum[((-1)^(1+sig)*(k*Sqrt[(k^2-(k0+I*c*sig)^2)/k^2]+(k0+I*c*sig)*ArcSin[(k0+I*c*sig)/k])*Binomial[kap,sig])/k0^3,{sig,0,kap}]
Nint30[kap_,c_,k0_,k_]:=NIntegrate[integrand[r,k,k0,c,kap,3,0],{r,0,Infinity}]
Lsum30relerror[kappa_,c_,k0_,k_]:=Abs[(NIntegrate[integrand[r,k,k0,c,kappa,3,0],{r,0,Infinity}]-Lsum30[kappa,c,k0,k])/Lsum30[kappa,c,k0,k]]

Lsum30relerror[6,0.1,1,1.1]

Lsum30relerror[2,0.1,1.8,0.7]

Lsum30[2,0.1,1.8,0.7]

Nint30[2,0.1,1.8,0.7]

Sum[((-1)^(1+sig)*(k*Sqrt[(k^2-(k0+I*c*sig)^2)/k^2]+(k0+I*c*sig)*ArcSin[(k0+I*c*sig)/k])*Binomial[kap,sig])/k0^3,{sig,0,kap}]

Lsum30alt[kap_,c_,k0_,k_]:=Sum[((-1)^(1+sig)*(k*Sqrt[(k^2-(k0+I*c*sig)^2)/k^2]+(k0+I*c*sig)*(ArcSin[(k0+I*c*sig)/k]+I*Pi))*Binomial[kap,sig])/k0^3,{sig,0,kap}]

Lsum30alt[2,0.1,1.8,0.7]