  We present the calculation of the order $\alpha_s^2$ corrections to the
coefficient functions contributing to the longitudinal ($F_L(x,Q^2)$) and
transverse fragmentation functions ($F_T(x,Q^2)$) measured in electron-positron
annihilation. The effect of these higher order QCD corrections on the behaviour
of the fragmentation functions and the corresponding longitudinal
($d\sigma_L(x,Q^2)/dx$) and transverse cross sections ($d\sigma_T(x,Q^2)/dx$)
are studied. In particular we investigate the dependence of the above
quantities on the mass factorization scale ($M$) and the various
parameterizations chosen for the parton fragmentation densities $D_p^H(x,M^2)$
($p=q,g$; $H=\pi^\pm, K^\pm, P, \bar{P}$). Our analysis reveals that the order
\alpha_s^2 contributions to $F_L(x,Q^2)$ are large whereas these contributions
to $F_T(x,Q^2)$ are small. From the above fragmentation functions one can also
compute the integrated cross sections $\sigma_L$ and $\sigma_T$ in an
independent way. The sum $\sigma_{tot} = \sigma_L + \sigma_T$, corrected up to
order \alphastwo, agrees with the well known result in the literature providing
us with an independent check an our calculations.
