The transverse, longitudinal and asymmetric components of the fragmentation function are measured from the inclusive charged particles in e⁺e⁻ collisions at LEP. The study of the process e⁺e⁻ → h + X provides a test of the QCD predictions using these fragmentation functions. The transverse σT and longitudinal σL components of the total hadronic cross section σtot are evaluated from the fragmentation functions. They are determined to be
respectively. The strong coupling constant is calculated from the longitudinal cross section in next-to-leading order of perturbative QCD, giving
Including non-perturbative power corrections leads to
By integrating the fragmentation functions, we determine the average charged particle multiplicity to be
From the longitudinal fragmentation function we are able to extract the gluon fragmentation function. The fragmentation functions in and light quark events are compared.