Becker, Becker and Schwarz in the book String theory and M-theory say on page 72 that $$\partial X^{\mu}(z)\partial X^{\nu}(w)=-\frac{1}{4}\frac{\eta^{\mu\nu}}{(z-w)^2}+\cdots $$ the OPE of two operators $\partial X^{\mu}(z)$ and $\partial X^{\nu}(w)$. I have some questions:
why does the author compute $\langle \partial X^{\mu}(z)\partial X^{\nu}(w)\rangle$?
why do $\langle \partial X^{\mu}(z)\partial X^{\nu}(w)\rangle$ and $: \partial X^{\mu}(z)\partial X^{\nu}(w):$ mean singular and non-singular parts of $\partial X^{\mu}(z)\partial X^{\nu}(w)$, respectively?
Why we compute $\langle \partial X^{\mu}(z)\partial X^{\nu}(w)\rangle$? or more operators. What does it tell us?
I think there is formula like $$OPE(AB)=\langle AB\rangle +:AB:$$ for two operators $A$ and $B$ that does not exist in the book. Is there such a formula in the physics?
