I will reformulate my question in the hopes it will be more intelligible.
First, I defined the functions:
f1[r_] = b*SphericalBesselJ[l, α*r]; f2[r_] = c*SphericalHankelH1[l, I*β*r]; g1[r_] := D[f1[r], r]/(f1[r]); g2[r_] := D[f2[r], r]/(f2[r]);
And I want to set:
Is it possible to use these two equations to find another equation that relates
Alpha, Beta and