Origin of the eq. 6.190 in the Di Francesco's book conformal field theory Recently I am reading Philippe Francesco's Book: Conformal Field Theory. Actually, I am having problems in understanding the origin of the following equation: 
$\mathscr{F}_{1}=\frac{\left(h_{p}+h_{2}-h_{1}\right)\left(h_{p}+h_{3}-h_{4}\right)}{2h_{p}}  .\tag{6.190} $ 
Comparing the equation 6.189 and 6.187, I suppose that
$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\mathscr{F}_{1}=\beta_{34}^{p\{1\}}\frac{\left\langle h_{1}\left|\phi_{2}\left(1\right)L_{-1}\right|h_{p}\right\rangle }{\left\langle h_{1}\left|\phi_{2}\left(1\right)\right|h_{p}\right\rangle } $
but I don't know what I need to do to arrive at the equation 6.190. I will be really grateful for your help. 
 A: Using the expression 
$$\mathscr{F}_{1}=\frac{\left\langle h_{1}\left|\phi_{2}\left(1\right)L_{-1}\right|h_{p}\right\rangle }{\left\langle h_{1}\left|\phi_{2}\left(1\right)\right|h_{p}\right\rangle }\beta_{34}^{p\{1\}}$$ 
for this purpose we use $\left\langle h_{1}\left|\phi_{2}\left(1\right)\right|h_{p}\right\rangle =\left(C_{p12}\right)^{1/2}$ 
and the commutation relation 
$\left[L_{n},\phi\left(w,\overline{w}\right)\right]=\left(h\left(n+1\right)w^{n}+w^{n+1}\partial\right)\phi\left(w,\overline{w}\right).\tag{6.28}$ 
with this equation is possible to rewrite 
$$\left\langle h_{1}\left|\phi_{2}\left(1\right)L_{-1}\right|h_{p}\right\rangle =\left\langle h_{1}\left|L_{-1}\phi_{2}\left(1\right)-\left[L_{-1},\phi_{2}\left(1\right)\right]\right|h_{p}\right\rangle$$
like $\left\langle h_{1}\right|L_{-1}=0$ (eq 7.10), we get 
$$\left\langle h_{1}\left|\phi_{2}\left(1\right)L_{-1}\right|h_{p}\right\rangle =\left\langle h_{1}\left|-\partial\phi_{2}\left(1\right)\right|h_{p}\right\rangle =-\left.\partial_{z}\left\langle h_{1}\left|\phi_{2}\left(z\right)\right|h_{p}\right\rangle \right|_{z=1}$$
in this point, we use the equation 
$$\left\langle \phi_{r}\left|\phi_{1}\left(z,\overline{z}\right)\right|\phi_{2}\right\rangle =\frac{C_{r12}}{z^{h_{1}+h_{2}-h_{r}}\overline{z}^{\overline{h_{1}}+\overline{h_{2}}-\overline{h_{r}}}}.\tag{6.166}$$
obtaining 
$$\left\langle h_{1}\left|\phi_{2}\left(1\right)L_{-1}\right|h_{p}\right\rangle =-\left.\partial_{z}\left\langle h_{1}\left|\phi_{2}\left(z\right)\right|h_{p}\right\rangle \right|_{z=1}=\left(h_{p}+h_{2}-h_{1}\right)\left(C_{12p}\right)^{1/2}$$
for this reason $\mathscr{F}_{1}=\left(h_{p}+h_{2}-h_{1}\right)\beta_{34}^{p\{1\}}$. Now is necessary to obtain the expression for $\beta_{34}^{p\{1\}}$ with $h_{3}\neq h_{4}$, It is possible reproducing the same steps from equation 6.169 to 6.175 with $h_{1}\neq h_{2}$ obtaining that the equation 6.175 is replaced by 
$$L_{n}\left|N+n,h_{p}\right\rangle =\left(h_{p}-h_{2}+nh_{1}+N\right)\left|N,h_{p}\right\rangle .\tag{6.175}$$
and finally
$$L_{1}\left|1,h_{p}\right\rangle =\left(h_{p}-h_{2}+h_{1}\right)\left|h_{p}\right\rangle =\beta_{12}^{p\{1\}}L_{1}L_{-1}\left|h_{p}\right\rangle =2h_{p}\beta_{12}^{p\{1\}}\left|h_{p}\right\rangle$$
 we arrive to 
$$\beta_{12}^{p\{1\}}=\frac{\left(h_{p}-h_{2}+h_{1}\right)}{2h_{p}}$$ 
using this expression with the indexes $12\rightarrow34$ we arrive in the equation 6.190.
