Finding OPE from conformal Ward identity WZW model

I’m working through section 15.1.3. of Di Francesco’s CFT textbook. I don’t understand the steps going between (15.42) and (15.43). They say to substitute
$delta_omega J = sum_{b,c} i f_{abc} omega^b J^c – k partial_z omega^a$ into the Ward identity
begin{equation}
delta_{omega, bar{omega}}leftlangle X rightrangle = -frac{1}{2pi i } oint dz sum_a omega^a leftlangle J^a X rightrangle + frac{1}{2pi i} oint sum_a bar{omega}^a leftlangle bar{J}^a X rightrangle
end{equation}
to get the OPE
begin{equation}
J^a(z) J^b (w) sim frac{k delta_{ab}}{(z-w^2)} + sum_c i f_{abc} frac{J^c(w)}{(z-w)}.
end{equation}
I know I need to use the residue theorem but I can’t see the steps I need to take to get to the OPE. Thanks very much for any help you can give me.