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We carry out a comprehensive study of the quark-to-meson fragmentationfunction in the 't Hooft model, i.e., the two-dimensional QuantumChromodynamics (QCD) in $N_c\to \infty$ limit, following the operatordefinition pioneered by Collins and Soper. We apply the Hamiltonian approach aswell as the diagrammatic approach to construct the functional form of thequark-to-meson fragmentation function in terms of the meson's light-cone wavefunction. For the sake of comparison, we also investigate the heavy quarkfragmentation into quarkonium in two-dimensional QCD within the framework ofthe nonrelativistic QCD (NRQCD) factorization, at the lowest order in quarkvelocity. In the heavy quark limit, the quark fragmentation function obtainedfrom the {\it ab initio} method agrees well, both analytically and numerically,with that obtained from the NRQCD approach. This agreement might be regarded asa nontrivial justification for the validity of both field-theoreticalapproaches to compute the heavy quark fragmentation function.

Heavy quark fragmentation function in 't Hooft Model