Simply Amusing Algebra and Analysis or Electromagnetic and Gravitational Fields in the Single System of Equations

Abstract: In this article the algebra and the basis of corresponding analysis in 4-dimensional spaces are constructed, in pseudoeuclidean with signature (1, -1, -1, -1) and pseudo-Riemannian corresponding to the real space-time. In both cases the analogues of Cauchy-Riemann conditions are obtained. They are the systems of 1-st order partial differential equations, linear for the pseudoeuclidean and quasi-linear for the pseudo-Riemannian space (linear as about the components of differentiable function ant its derivatives so about the derivatives of metric tensor). The general solution for pseudoeuclidean space which is the flat waves of components of dependent function, and special (spherical-symmetric) wave-like (as for the components of differentiable function so for the components of metric tensor) solution for the pseudo-Riemannian space are got. In the last case the absence of central singularity for the components of metric tensor is interesting. From the Cauchy-Riemann condition follows that the differentiable function is constant along some isotropic curves given by 1-st order differential equations. The demand these curves to be geodetic lines leads to the differential restrictions for the metric tensor itself. The special kind of these restrictions is obtained. The hypothesis that the differentiable function can be interpreted as an electromagnetic field is expressed.