\newcommand{\mr}{\mathrm}
\newcommand{\ff}{fragmentation function}
\newcommand{\ffs}{fragmentation functions}
\newcommand{\ee}{e$^+$e$^-$}
\newcommand{\Qj}{$Q_{\mr jet}$}
\newcommand{\xe}{$x_{\mr E}$}
\newcommand {\egstar} { E_{\mathrm g}^* }


Fragmentation functions of quark and gluon jets as measured by OPAL.

Scaling violations of quark and gluon jet \ffs\ are studied in \ee\
annihilations at $\sqrt s =$ 91.2 and 183--209~GeV using data collected
with the OPAL detector at LEP. The scale dependence of the flavour inclusive,
udsc and b \ffs\ from unbiased jets is measured at $\sqrt s/2=$ 45.6 and
91.5--104.5~GeV. Biased jets are used to extract the flavour inclusive, udsc
and b, and gluon \ffs\ in the ranges $Q_{\mr jet}=$ 4--42, 4--105 and
4--70~GeV, respectively, where \Qj\ is the jet energy scale. Three methods
are used to extract the \ffs, namely the b-tag and energy-ordering methods
for biased jets, and the hemisphere method for unbiased jets. The results
obtained using these methods are found to be consistent with each other.
The udsc jet results above the scale of 45.6~GeV, the gluon jet results above
30~GeV (except for the scale of 40.1~GeV), and the b jet results at all scales
except 45.6~GeV represent new measurements. The results of this analysis are
compared with existing lower energy \ee\ data and with previous
results from DELPHI and OPAL. The overall consistency of the biased jet
results with the unbiased jet results suggests that \Qj\ is a generally
appropriate scale in events with a general three-jet topology.
The scaling violation is observed to be positive for lower \xe\ and
negative for higher \xe, for all the types of \ffs. The gluon jet \ff\ exhibits
stronger scaling violation than that of udsc jets.\\

The bias of the procedure used to construct biased jet \ffs\ is estimated by
studying hadron level Monte Carlo generator events. In explaining the
observed differences between biased and unbiased jet results, we note the
effects of non-negligible masses of hadrons and b-quarks at low scales.
Due to the considerable bias found for the gluon jet \ffs\ in the region of
$x_{\mr E} > 0.6$, precautions should be taken when comparing the biased gluon
jet results with theory.\\

The data are compared to the predictions of NLO calculations. In a wide
range of \xe, all calculations satisfactorily describe the
data for the udsc jet \ffs.
The description is worse and the spread between the predictions larger for the
b and gluon jet \ffs, in particular in regions of very low and high \xe. \\

The data are also compared with predictions of three Monte Carlo models,
PYTHIA 6.125, HERWIG 6.2 and ARIADNE 4.08.
A reasonable agreement with data is observed for all models, except for high
\xe\ region with small scales ($\lesssim$ 14~GeV) in case of the udsc
and gluon jet \ffs.\\

The charged particle multiplicities of udsc, b and inclusive hadronic
events are obtained by integrating the measured \ffs. All values are
found to be in agreement with previous measurements, where available.\\


The first experimental study to use the jet boost algorithm,
a method based on the QCD dipole model to extract properties of
unbiased gluon jets from
{\ee}$\,\rightarrow\,$$\mathrm q\overline{q}g$ events, has also
been presented. We test the jet boost algorithm using the Herwig Monte 
Carlo QCD simulation program, comparing the results of this method to
those derived from unbiased gluon jets defined by hemispheres
of inclusive gg events from a color singlet point source.
We find that the results of the jet boost algorithm
for the multiplicity distribution are in close
correspondence to those of the gg hemispheres for jet energies
$\egstar$ larger than about 5~GeV.
For the fragmentation functions,
the results of the two methods agree to good precision
for $\egstar$$\,\gtsim\,$14~GeV.
Therefore, the fragmentation function of unbiased gluon jets
have been measured only at two points, namely at 14.24 and 17.72~GeV.