We start from the basic formula from the theory of radiative transfer, namely the formal solution of the radiative transfer equation,
If we now consider a CONSTANT SOURCE FUNCTION , then Eq.(2) gives the solution
In practically observational view, the relation between monochromatic specific intensity and a Plank Function at is defined using emissivity or , namely . In the same way, we consider the Rayleigh-Jeans Regime, the opacity will be . Then we often use an exponential function to fit the data as
And , is the frequency where . is the so-called emissivity index, which is about 1.5~2, see L. Dunne et al., 2001 (1). For big dust grains, it tends to be 2 while for smaller ones, the value becomes lower.
If we assume is a constant along the path, then we can reform Eq.(8) as
When the redshift is non-negligible, we need to use luminosity distance to replace the distance used above. Thus the flux and luminosity should be corrected (called K-correction). Because the energy of the source is constant. So we use
The left one above is the formula (1) in Beelen et al., 2006 (3). IF we want to get the far-infrared (FIR hereafter) luminosity, we should simply integrate over the FIR (cover the whole dust continuum) as,
This equation (actually this equation contains the K-correction as the is defined in the way above) is the Eq. (2) in Andrew B. Blain et al., 2002 (2).
Additionally, we can use Eq. (9) to reform the monochromatic flux:
We can also derive the relation between opacity and mass-absorption coefficient, which in fact can be rewritten from Eq.(8) as: , as below
This formula is often used to calculate the dust mass from FIR/submm radiation.
 L. Dunne and S. Eales, 2001, MNRAS, 327, 697,
 Blain, A. W., Smail, I., Ivison, R. J., Kneib, J.P., Frayer, D. T. 2002, Phys. Rep., 369, 111.
 Beelen, A., Cox, P., Benford, D. J., et al. 2006, ApJ, 642, 694.