A measurement of the $Z(\rightarrow\mu^+\mu^-)$+jet production cross-section in $pp$ collisions at a centre-of-mass energy $\sqrt{s} = 7$ TeV is presented. The analysis is based on an integrated luminosity of $1.0\,\text{fb}^{-1}$ recorded by the LHCb experiment. Results are shown with two jet transverse momentum thresholds, 10 and 20 GeV, for both the overall cross-section within the fiducial volume, and for six differential cross-section measurements. The fiducial volume requires that both the jet and the muons from the Z boson decay are produced in the forward direction ($2.0<\eta<4.5$). The results show good agreement with theoretical predictions at the second-order expansion in the coupling of the strong interaction.
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