I have questions regarding the equation (2.2.4) in Polchinski Vol 1:
$$ X^\mu (z_1,\bar{z}_1) X^\nu(z_2,\bar{z}_2) = -\frac{\alpha'}{2}\eta^{\mu\nu} \ln|z_{12}|^2 + \sum_{k=1}^\infty \frac{1}{k!}\left[(z_{12})^k:X^\nu \partial^k X^\mu(z_2,\bar{z}_2): + (\bar{z}_{12})^k:X^\nu \bar{\partial}^k X^\mu(z_2,\bar{z}_2):\right]. \tag{2.2.4}$$
Here $z_{12} = z_1-z_2$, and the :: means normal-ordering:
$$ :X^\mu (z_1,\bar{z}_1) X^\nu(z_2,\bar{z}_2): = X^\mu (z_1,\bar{z}_1) X^\nu(z_2,\bar{z}_2) + \frac{\alpha'}{2}\eta^{\mu\nu} \ln|z_{12}|^2 .\tag{2.1.21b}$$
Now, how can we accommodate the normal-ordered string $$:X^\nu \partial^k X^\mu(z_2,\bar{z}_2):$$ in the definition above? I can imagine there are at least two complications:
a derivative is involved, and
the product product is now at one single point $(z_2,\bar{z}_2)$, where would the log term $\ln|z_{12}|^2$ go?