Consider a source observed to have apparent magnitude
*m*_{R} when observed through photometric bandpass
*R*, for which one wishes to know its absolute magnitude
*M*_{Q} in emitted-frame bandpass *Q*. The
*K* correction *K*_{QR} for this source is
*defined* by

(2) |

where *DM* is the distance modulus, defined by

(3) |

where *D*_{L} is the luminosity distance (*e.g.*
Hogg 1999)
and 1 pc = 3.086 × 10^{16} m.

The apparent magnitude *m*_{R} of the source is related to
its spectral density of flux
*f*_{}() (energy per unit time per unit
area per unit frequency) by

(4) |

where the integrals are over the observed frequencies
_{0};
*g*^{R}_{}() is the spectral density of flux
for the zero-magnitude or "standard" source, which, for Vega-relative
magnitudes, is Vega (or perhaps a weighted sum of a certain set of A0
stars), and, for AB magnitudes
(Oke & Gunn 1983),
is a hypothetical constant source with
*gAB*_{}() = 3631 Jy (where
1 Jy = 10^{-26} W m^{-2} Hz^{-1} =
10^{-23} erg cm^{-2} s^{-1} Hz^{-1}) at
all frequencies
; and
*R*() describes the
bandpass, as follows:

The value of *R*() at
each freqency
is the mean contribution
of a photon of frequency
to the output signal from the
detector. If the detector is a photon counter, like a CCD, then
*R*() is just
the probability that a photon of frequency
_{0} gets counted. If
the detector is a bolometer or calorimeter, then
*R*() is the energy
deposition *h* per
photon times the fraction of photons of
energy that get absorbed into
the detector. If
*R*() has
been properly computed, there is no need to write different integrals
for photon counters and bolometers. Note that there is an implicit
assumption here that detector nonlinearities have been corrected.

The absolute magnitude *M*_{Q} is defined to be the
apparent magnitude that the source *would* have if it were
10 pc away, at
rest (*i.e.*, not redshifted), and compact. It is related to the
spectral density of the luminosity
*L*_{}() (energy per unit
time per unit frequency) of the source by

(5) |

where the integrals are over emitted (*i.e.*, rest-frame) frequencies
_{e},
*D*_{L} is the luminosity distance, and
*Q*() is the
equivalent of *R*() but
for the bandpass *Q*. As mentioned above,
this does not require *Q* = *R*, so this will lead to,
technically, a
generalization of the *K* correction. In addition, since the
*Q* and
*R* bands can be zero-pointed to different standard sources
(*e.g.*, if
*R* is Vega-relative and *Q* is AB), it is not necessary that
*g*^{Q}_{} =
*g*^{R}_{}.

If the source is at redshift *z*, then its luminosity is related to
its flux by

(6) |

(7) |

The factor of (1 + *z*) in the luminosity expression
(6) accounts for the fact that the flux and
luminosity are not bolometric but densities per unit freqency. The
factor would appear in the numerator if the expression related flux
and luminosity densities per unit wavelength.

Equation (2) holds if the *K* correction *K*_{QR} is

(8) |

Equation (8) can be taken to be an operational
definition, therefore, of the *K* correction, from observations through
bandpass *R* of a source whose absolute magnitude
*M*_{Q} through
bandpass *Q* is desired. Note that if the *R* and *Q*
have different zero-point definitions, the
*g*^{R}_{}(_{e}) in the numerator will
be a different function from the
*g*^{Q}_{}(_{0})
in the denominator.

In equation (8), the *K* correction was defined in
terms of the apparent flux
*f*_{}() in the observed frame. This
is the direct observable. Most past discussions of the *K*
correction (*e.g.*
Oke & Sandage 1968;
Kim et al. 1996)
write equations for the *K* correction in
terms of either the flux or luminosity in the emitted frame.
Transformation from observed-frame flux
*f*_{}(_{0}) to
emitted-frame luminosity
*L*_{}(_{e}) gives

(9) |

In the above, all calculations were performed in frequency units. In
wavelength units, the spectral density of flux
*f*_{}() per unit
frequency is replaced with the spectral density of flux
*f*_{}()
per unit wavelength using

(10) |

(11) |

where *c* is the speed of light. The *K* correction becomes

(12) |

where, again,
*R*() is defined
to be the mean contribution to
the detector signal in the *R* bandpass for a photon of wavelength
and
*Q*() is defined
similarly. Note that the
hypothetical standard source for the AB magnitude system, with
*gAB*_{}() constant, has
*gAB*_{}()
not constant but rather
*gAB*_{}()
= *c*
^{-2}
*gAB*_{}().

Again, transformation from observed-frame flux
*f*_{}(_{o}) to emitted-frame luminosity
*L*_{}(_{e}) gives

(13) |

Equation (13) becomes identical to the equation for *K* in
Oke & Sandage (1968)
if it is assumed that *Q* = *R*, that
*g*^{Q}_{} =
*g*^{R}_{},
that the variables
_{0},
*F*(), and
*S*_{i}()
in Oke & Sandage (1968)
are set to

(14) |

and that the integrand
is used differently
in each of the two integrals. Similar transformations make the equations
here consistent with those of
Kim et al. (1996),
although they distinguish
between the classical *K* correction and one computed for photon
counting devices (an unnecessary distinction); their most similar
equation is that given for *K*_{xy}^{counts}.