Why should a holomorphic function be expanded in Laurent series rather than Taylor series? In 2d free conformal field  theory, there is an operator equation:
$$
\partial\bar\partial\hat{X}^\mu\left(z,\bar z\right)=0
$$
Why can it have Laurent expansion like this below rather than Taylor
$$
\partial X^\mu\left(z\right)=-i\left(\frac{\alpha'}{2}\right)^{1/2}\sum_{m=-\infty}^{\infty}\frac{\alpha_m^\mu}{z^{m+1}}, \\
\bar\partial X^\mu\left(\bar z\right)=-i\left(\frac{\alpha'}{2}\right)^{1/2}\sum_{m=-\infty}^{\infty}\frac{\tilde{\alpha}_m^\mu}{\bar z^{m+1}}.
$$
 A: I) The string (target space) coordinates 
$X^{\mu}(\tau_E,\sigma)$ 
depend on a (world sheet) spatial coordinate $\sigma$ and a (world sheet) temporal coordinate $\tau_E$ (which we here have Wick-rotated to Euclidean time, hence the subscript $E$). 
II) Similar to how one quantizes a field in QFT, in string theory, the Fourier series expansion of string coordinates $X^{\mu}$ will consists of creation and annihilation parts. 
III) The (world sheet) coordinates $(\tau_E,\sigma)$ may be organized into a complex coordinate 
$$z=e^{\tau_E+i\sigma}.$$ 
Equal-time contours corresponds to concentric circles in the complex $z$-plane.
The origin $z=0$ corresponds to the infinite past, while the infinite future corresponds to large $|z|=\infty$. This punctured complex $z$-plane $\mathbb{C}\backslash\{0\}$ sets the stage for Laurent series (and radial ordering prescription for operators).
IV) In the Fourier series expansion of the string $X^{\mu}(z,\bar{z})$, the creation operator modes become multiplied with $z$ and $\bar{z}$ powers, while the annihilation operator modes become multiplied with $\frac{1}{z}$ and $\frac{1}{\bar{z}}$powers. In this way the string $X^{\mu}(z,\bar{z})$ becomes a Laurent series.
