  We estimate the power corrections (infrared renormalon contributions) to the
coefficient functions for the transverse, longitudinal and asymmetric
fragmentation functions in $e^+e^-$ annihilation, using a method based on the
analysis of one-loop Feynman graphs containing a massive gluon. The leading
corrections have the expected $1/Q^2$ behaviour, but the gluonic coefficients
of the longitudinal and transverse contributions separately have strong
singularities at small $x$, which cancel in their sum. This leads to $1/Q$
corrections to the longitudinal and transverse parts of the annihilation cross
section, which cancel in the total cross section.
