Numerical method for finding the wavefunction from the wave matrix in scattering problem

I am trying to solve the last Exploration Problem from chapter 26 of the book 'Computational physics: problem solving with python' R. Landau, third edition. Basically what I need is to find the reduced wavefunction $u(r)$ with $l=0$, for a local Dirac delta potential.

The problem says that there is a ${F}^{-1}$ matrix (supposedly called the wave matrix) that can be used to find the wavefunction, via the relation: $$u(r)=N_0\sum_{i=1}^N\frac{\sin(k_ir)}{k_ir}F^{-1}(k_i,k_0),$$ I just wonder if that last expression is correct, I have tried to find it in other literature, but I have not been able to. What I ask for is a reference of this equation from a book or an article that proves this relation. I need it because when I apply the equation the wavefunction does not behave as it should analytically (sine waves); so I may have an error on my code (which I think is not the case this time because every other quantity in my scattering project did behave according to the analytic expressions), or the expression is incorrect (the author may have a typo), or I interpret it incorrectly.

The $F$ matrix comes from the equation: $${F}R=V$$

where $R$ is the scattering amplitude in momentum space, solved from the Lippmann-Schwinger equation: $$R(k,k')=V(k,k')+\frac{2}{\pi}\mathcal{P}\int_0^\infty dp\frac{p^2V(k',p)R(p,k)}{(k_0^2-p^2)/(2\mu)}.$$ and the potential in the momentum space is given by $$V(k,k')=\frac{1}{kk'}\int_0^\infty dr\: \sin(kr)V(r)\sin(k'r).$$