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Spatial Complex Variables and Applications

Shanguang Tan
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Abstract

We introduce a new kind of numbers $s=e_{xy}(x+iy)+z$ called Spatial Complex Numbers and establish a new theory of Spatial Complex Variables here where $e_{xy}$ denotes the $xy$ coordinate plane. Studies of the theory of spatial complex variables include: (1) Definition and Algebras; (2) Analytic functions, Cauchy-Riemann equations, and spatial harmonic functions; (3) Elementary functions; (4) Spatial contour integrals; (5) Surface integrals; (6) Taylor and Laurent series; (7) Residue theorem; (8) Conformal mapping; (9) Applications of three-dimensional fluid flow and Navier-Stokes equations; (10) The Schwarz-Christoffel Transformation; (11) Spatial circle and spherical surface integral formulas of the Poisson type; (12) Applications of Residues in two-dimensional improper integrals. Results show that spatial complex numbers are with good properties of fields just as the two-dimensional complex numbers. Spatial complex numbers follow the commutative and associative laws of multiplication of numbers, the distributive law of addition and multiplication of numbers, and other laws of numbers. The theory of spatial complex variables can be used to solve many three-dimensional problems in sciences and technologies, such as expressed by the three-dimensional Laplace equation. So numbers are developed from the real and two-dimensional complex numbers to three-dimensional complex numbers, and the number system is enlarged.

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Shanguang Tan. 2007-03-23. Spatial Complex Variables and Applications. https://arxiv.org/abs/math/0703688

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