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The fragmentation of a light parton i to a jet containing a light energetichadron h, where the momentum fraction of this hadron as well as the invariantmass of the jet is measured, is described by "fragmenting jet functions". Wecalculate the one-loop matching coefficients J_{ij} that relate the fragmentingjet functions G_i^h to the standard, unpolarized fragmentation functions D_j^hfor quark and gluon jets. We perform this calculation using various IRregulators and show explicitly how the IR divergences cancel in the matching.We derive the relationship between the coefficients J_{ij} and the quark andgluon jet functions. This provides a cross-check of our results. As anapplication we study the process e+ e- to X pi+ on the Upsilon(4S) resonancewhere we measure the momentum fraction of the pi+ and restrict to the dijetlimit by imposing a cut on thrust T. In our analysis we sum the logarithms oftau=1-T in the cross section to next-to-next-to-leading-logarithmic accuracy(NNLL). We find that including contributions up to NNLL (or NLO) can have alarge impact on extracting fragmentation functions from e+ e- to dijet + h.

Parton Fragmentation within an Identified Jet at NNLL