We investigate the anisotropic clustering of the Baryon Oscillation Spectroscopic Survey (BOSS) Data Release 12 (DR12) sample, which consists of $1,198,006$ galaxies in the redshift range $0.2 < z < 0.75$ and a sky coverage of $10,252,$deg$^2$. We analyse this dataset in Fourier space, using the power spectrum multipoles to measure Redshift-Space Distortions (RSD) simultaneously with the Alcock-Paczynski (AP) effect and the Baryon Acoustic Oscillation (BAO) scale. We include the power spectrum monopole, quadrupole and hexadecapole in our analysis and compare our measurements with a perturbation theory based model, while properly accounting for the survey window function. To evaluate the reliability of our analysis pipeline we participate in a mock challenge, which resulted in systematic uncertainties significantly smaller than the statistical uncertainties. While the high-redshift constraint on $fsigma_8$ at $z_{rm eff}=0.61$ indicates a small ($sim 1.4sigma$) deviation from the prediction of the Planck $Lambda$CDM model, the low-redshift constraint is in good agreement with Planck $Lambda$CDM. This paper is part of a set that analyses the final galaxy clustering dataset from BOSS. The measurements and likelihoods presented here are combined with others in~citet{Alam2016} to produce the final cosmological constraints from BOSS.
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