Commutative complex numbers in four dimensions
  Commutative complex numbers of the form u=x+\alpha y+\beta z+\gamma t in 4 dimensions are studied, the variables x, y, z and t being real numbers. Four distinct types of multiplication rules for the complex bases \alpha, \beta and \gamma are investigated, which correspond to hypercomplex entities called in this paper circular fourcomplex numbers, hyperbolic fourcomplex numbers, planar fourcomplex numbers, and polar fourcompex numbers. Exponential and trigonometric forms for the fourcomplex numbers are given in all these cases. Expressions are given for the elementary functions of the fourcomplex variables mentioned above. Relations of equality exist between the partial derivatives of the real components of the functions of fourcomplex variables. The integral of a fourcomplex function between two points is independent of the path connecting the points. The concepts of poles and residues can be introduced for the circular, planar, and polar fourcomplex numbers, for which the exponential forms depend on cyclic variables. A hypercomplex polynomial can be written as a product of linear factors for circular and planar fourcomplex numbers, and as a product of linear or quadratic factors for the hyperbolic and polar fourcomplex numbers. 
Silviu Olariu (National Institute of Physics and Nuclear Engineering,   Tandem Laboratory, Magurele, Bucharest, Romania)