Jon,
Oh, yes! So, let's cover this point as well. The topic began as a discussion on spectra. If it is relative or uncalibrated, each pixel's Fv in the spectrum will be an integral over the wavelengths range within the pixel, as you specified. A calibration COULD remove this and give a true F_nu. Whether it really does or not depends on how sophisticated the calibration code is, but generally the difference between the two in a spectrum is ignorable.
In photometry, if one bans magnitudes, one has two types of F_nu. In one, the filter's transmission curve has been theoretically removed, perhaps by assuming a spectral template based on the type or temperature of the object. Here one has to be alert to the fact that this is always just a crude estimation. The other is a band-F_nu which is actually the numerator in your expression below, ie what you get from band-F_nu(std)*10^(-m/2.5).
The convention has been to express the former as Flux F_nu and the latter is just kept as a magnitude M_band. But not always so, and there is no critical argument that says it must remain so. It is after all a matter of taste. More properly, it is a matter of how distasteful it is to provide some additional code to handle astronomers' quirks. I think the two of us agree that a bit of extra code is not so terrible.
Ed
PS - I think with this, I sign off, as I am no longer vo funded.
Jonathan McDowell wrote:
> Ed, I respectfully disagree.
> The definition of mag is actually
> -2.5 log ( \int Ftarg(lambda)T(lambda)dlambda / \int Fstd(lambda)T(lambda)dlambda )
> where T is the transmission curve which can be quite broad.
> This is important, because the inversion depends on the spectrum of the source
> ("color correction") and therefore if you pick a simple conversion and just
> give the number in Jy, you have lost information. So it's more than just
> the compression efficiency that you describe.
> - Jonathan