J[l_,k_,f_,sigma_]= l* JacobiCN[Sqrt[(l+k)/(4*sigma)]*f,Sqrt[l/(l+k)]]

l JacobiCN[(1/2) f (Sqrt[((k+l)/sigma)]),(Sqrt[(l/(k+l))])]

Manipulate[Plot[J[l,k,f,sigma],{f,0,100}],{sigma,.1,10},{k,0,5},{l,0.1
,5}]