  In our joint papers [FL1-FL2] we revive quaternionic analysis and show deep
relations between quaternionic analysis, representation theory and
four-dimensional physics. As a guiding principle we use representation theory
of various real forms of the conformal group. We demonstrate that the
requirement of unitarity of representations naturally leads us to the
extensions of the Cauchy-Fueter and Poisson formulas to the Minkowski space,
which can be viewed as another real form of quaternions. However, the Minkowski
space formulation also brings some technical difficulties related to the fact
that the singularities of the kernels in these integral formulas are now
concentrated on the light cone instead of just a single point in the initial
quaternionic picture. But the same phenomenon occurs when one passes from the
complex numbers to the split complex numbers (or hyperbolic algebra). So, as a
warm-up example we proved an analogue of the Cauchy integral formula for the
split complex numbers. On the other hand, there seems to be sufficient interest
in such formula among physicists. For example, see [KS] and the references
therein.
