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In this article the algebra and the basis of corresponding analysis in4-dimensional spaces are constructed, in pseudoeuclidean with signature (1, -1,-1, -1) and pseudo-Riemannian corresponding to the real space-time. In bothcases the analogues of Cauchy-Riemann conditions are obtained. They are thesystems of 1-st order partial differential equations, linear for thepseudoeuclidean and quasi-linear for the pseudo-Riemannian space (linear asabout the components of differentiable function ant its derivatives so aboutthe derivatives of metric tensor). The general solution for pseudoeuclideanspace which is the flat waves of components of dependent function, and special(spherical-symmetric) wave-like (as for the components of differentiablefunction so for the components of metric tensor) solution for thepseudo-Riemannian space are got. In the last case the absence of centralsingularity for the components of metric tensor is interesting. From theCauchy-Riemann condition follows that the differentiable function is constantalong some isotropic curves given by 1-st order differential equations. Thedemand these curves to be geodetic lines leads to the differential restrictionsfor the metric tensor itself. The special kind of these restrictions isobtained. The hypothesis that the differentiable function can be interpreted asan electromagnetic field is expressed.

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