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We aim to study the statistical properties of dusty star-forming galaxies,such as their number counts, luminosity functions (LF) and dust-obscuredstar-formation rate density (SFRD). We use state-of-the-art de-blended Herschelcatalogue in the COSMOS field, generated by combining the Bayesian sourceextraction tool XID+ and informative prior on the spectral energydistributions, to measure the number counts and LFs at far-infrared (FIR) andsub-millimetre (sub-mm) wavelengths.  Thanks to our de-confusion technique and deep multi-wavelength photometry, weare able to achieve more accurate measurements while probing ten times belowthe confusion limit. Our number counts at 250 microns agree well with previousHerschel studies. However, our counts at 350 and 500 microns are considerablybelow previous Herschel results. This is due to previous studies suffering fromsource confusion which worsens towards longer wavelength. Our number counts at450 and 870 microns show excellent agreement with previous determinationsderived from single dish and interferometric observations. Our measurements ofthe monochromatic LF and the total IR LF agree well with previous results. Theincreased dynamic range of our measurements allows us to better measure thefaint-end slope of the LF and measure the dust-obscured SFRD out to z~6. Wefind that the fraction of dust obscured star-formation activity is at itshighest around z~1 which then decreases towards both low and high redshift. Wedo not find a shift of balance between z~3 and z~4 in the cosmic star-formationhistory from being dominated by unobscured star formation at higher redshift toobscured star formation at lower redshift. However, we do find 3<z<4 to be aninteresting transition period as the fraction of the total SFRD that isobscured by dust is significantly lower at higher redshifts.

A multi-wavelength de-blended Herschel view of the statistical properties of dusty star-forming galaxies across cosmic time