VEGA System:

  magVega(Obj) = -2.5 x log(∫{Fl(Obj) x Sl x dl} / ∫{Fl(Vega) x Sl x dl})

Which can also be expressed as:
 
  magVega(Obj) = -2.5 x log(∫{Fν(Obj) x Sν x dν} / ∫{Fν(Vega) x Sν x dν})


Where:

  l        is wavelenght (λ); 
  ν        is frequency; 
  Fν(Obj)  is the object flux in ergs s-1cm-2Hz-1;
  Sν       is the instrumental (filter+CCD+telescope) response; and 
  Fν(Vega) is Vega's flux in ergs s-1cm-2Hz-1;

Meaning that by definition, Vega's magnitudes are 0.0 in all filters. However, because of uncertainties in the absolute flux calibration of Vega, magnitudes of this star have been slightly corrected over time.

AB System:

  magAB = -2.5 x log(∫{Fν(Obj) x Sν x dν} / ∫{Sν x dν}) - 48.6 


So that an object with Fν=constant (flat energy distribution) 
has the same magnitude in all bands, and all colors=0. 

AB &harr VEGA System Conversion:

From the definition of AB magnitudes:

  magAB(Obj) = -2.5 x log(∫{Fν(Obj) x Sν x dν} / ∫{Sν x dν}) - 48.6 

             = -2.5 x log(∫{Fν(Obj) x Sν x dν} x ∫{Fν(Vega) x Sν x dν} / ∫{Sν x dν} x ∫{Fν(Vega) x Sν x dν} ) - 48.6 

	     = magVega(Obj) + magAB(Vega) 


and the conversion between systems is simply given by the AB magnitude of Vega:


  conv = magAB(Vega) = -2.5 x log(∫{Fν(Vega) x Sν x dν} / ∫{Sν x dν}) - 48.6

Conversion factors for the DLS (and a few other) passbands:

                                           		 
     Deep Lens Survey     Landolt92 filters    SDSS filters	                    
     -----------------------------------------------------------
        		  LandoltU     1.00    sdssu        0.96
     DLSB        -0.09    LandoltB    -0.09    sdssg       -0.09
     DLSV         0.00    LandoltV    -0.00  	   		 
     DLSR         0.20    LandoltR     0.18    sdssr        0.16
     DLSI         0.45    LandoltI     0.46    sdssi        0.39
     DLSz         0.54			       sdssz        0.54

The conversion factors were computed from this flux calibrated Vega SED (1993 Kurucz models). Filters (already convolved with CCD-QE) used were: DLS BVRIz; Landolt 1992 UBVRI; and SDSS ugriz.