  We describe an explicit connection between solutions to equations $Df=0$ (the
Generalized Cauchy-Riemann equation) and $(D+M)f=0$, where operators $D$ and
$M$ commute. The described connection allows to construct a ``function theory''
(the Cauchy theorem, the Cauchy integral, the Taylor and Laurent series etc.)
for solutions of the second equation from the known function theory for
solution of the first (generalized Cauchy-Riemann) equation. As well known,
many physical equations related to the orthogonal group of rotations or the
Lorentz group (the Dirac equation, the Maxwell equation etc.) can be naturally
formulated in terms of the Clifford algebra. For them our approach gives an
explicit connection between solutions with zero and non-zero mass (or external
fields) and provides with a family of formulas for calculations.
\keywords{Dirac equation with mass, Clifford analysis.} \AMSMSC{30G35}{34L40,
81Q05}
