By Silviu Olariu (Eds.)

ISBN-10: 0444511237

ISBN-13: 9780444511232

Targeted platforms of hypercomplex numbers in n dimensions are brought during this e-book, for which the multiplication is associative and commutative, and that are wealthy adequate in houses such that exponential and trigonometric kinds exist and the innovations of analytic n-complex functionality, contour integration and residue might be defined.The booklet offers a close research of the hypercomplex numbers in 2, three and four dimensions, then provides the houses of hypercomplex numbers in five and six dimensions, and it keeps with an in depth research of polar and planar hypercomplex numbers in n dimensions. The essence of this ebook is the interaction among the algebraic, the geometric and the analytic aspects of the family.

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**Extra resources for Complex Numbers in Dimensions**

**Example text**

99), the components P, Q of an analytic function of twocomplex variable are solutions of the wave equation with respect to the variables x, y. 7 Integrals of twocomplex functions The singularities of twocomplex functions arise from terms of the form l/(u — uo)^\ with m > 0. Functions containing such terms are singular not only at u = no, but also at all points of the lines passing through no and which are parallel to the nodal lines. The integral of a twocomplex function between two points A, B along a path situated in a region free of singularities is independent of path, which means that the integral of an analytic function along a loop situated in a region free from singularities is zero,

2, where the plane through the point P and perpendicular to the trisector line (t) intersects the x axis at point A of coordinates (x + y + 2:, 0,0), the y axis at point B of coordinates (0, X + y + 2:, 0), and the z axis at point C of coordinates (0,0, x + y + 2). The azimuthal angle (j) of the tricomplex number x H- /iy + kz is defined as the angle in the plane 11 of the projection of P on this plane, measured from the line of intersection of the plane determined by the line {t) and the X axis with the plane 11, 0 < 0 < 27r.

44) + y^/7\ + • • •, px y = y V2! + y^bl + / / 8 ! + • • •. 45) From the series definitions it can be seen that ex 0 = 1, mx 0 = 0, px 0 = 0. The tridimensional polar cosexponential functions belong to the class of the polar n-dimensional cosexponential functions gnki and ex = 930, mx = iQ'aiiPx = fl32- It can be checked that cxy -\-pxy + mx y = exp y. 48) px (y + 2:) = mxy mxz + cxy px z -{-pxy ex z. 49) For y = z, Eqs. 51) px 2y = mx^ y + 2 ex y px z. 52) The cosexponential functions are neither even nor odd functions.

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