A measurement of the inclusive $W to mu u$ production cross-section using data from $pp$ collisions at a centre-of-mass energy of $sqrt{s} = 7 {rm , TeV}$ is presented. The analysis is based on an integrated luminosity of about $1.0 {rm , fb}^{-1}$ recorded with the LHCb detector. Results are reported for muons with a transverse momentum greater than $20 {rm , Gekern -0.1em V!/}c$ and pseudorapidity between 2.0 and 4.5. The $W^{+}$ and $W^{-}$ production cross-sections are measured to be $$sigma_{W^{+} to mu^{+} u} = 861.0 pm 2.0 pm 11.2 pm 14.7 {rm , pb},$$ $$sigma_{W^{-} to mu^{-}overline{ u}} = 675.8 pm 1.9 pm 8.8 pm 11.6 {rm , pb},$$ where the first uncertainty is statistical, the second is systematic and the third is due to the luminosity determination. Cross-section ratios and differential distributions as functions of the muon pseudorapidity are also presented. The ratio of $W^{+}$ to $W^{-}$ cross-sections in the same fiducial kinematic region is determined to be $$frac{sigma_{W^{+} to mu^{+} u}}{sigma_{W^{-} to mu^{-}overline{ u}}} = 1.274 pm 0.005 pm 0.009,$$ where the uncertainties are statistical and systematic, respectively. Results are in good agreement with theoretical predictions at next-to-next-to-leading order in perturbative quantum chromodynamics.
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