Download Complex Numbers in Dimensions by Silviu Olariu (Eds.) PDF

By Silviu Olariu (Eds.)

Designated structures 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 homes such that exponential and trigonometric kinds exist and the ideas of analytic n-complex functionality, contour integration and residue will be defined.The ebook 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 a close research of polar and planar hypercomplex numbers in n dimensions. The essence of this e-book is the interaction among the algebraic, the geometric and the analytic elements of the relatives.

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These expressions can be obtained by calculating first [h + fc)^ and [h - kf^. 57) where m is a natural number. 58) As a corollary, the following identities can be obtained from Eq. 58) by writing e^^'^^^y = e^ye^y and expressing e^y and e^y in terms of cosexponential functions via Eqs. 59) ex y mx y '\- cxy px y -f- mx y px y = —-e~2/ -f -e^y. 60) From Eqs. 60) it results that cx^ y 4- mx^ y + px^ y —exy mxy — cxy pxy — mxy pxy = exp(—y). 61) Then from Eqs. 61) it follows that cx*^ y 4- mx*^ y + px'^ y — 3cx y mx y px y = 1.

5 35 Elementary functions of a tricomplex variable It can be shown with the aid of Eq. 93) which are vaUd for real values of m. Thus Eqs. 93) define the power function u'^ of the tricomplex variable u. The power function is multivalued unless m is an integer. It can be inferred from Eq. 86) that, for integer values of m, {uu'r = u"^ w'"^. 94) For natural m, Eq. 20). For example if m = 2, it can be checked that the right-hand side ofEq. 93) is equal to (a:-f % + A:2:)2 = x'^-^2yz + h(z'^-\-2xz) + k{y'^+2xz).

If u = X + hy + kz^ then expw can be calculated as expw = exp:r • exp(/iy) • exp(/c2:). According to Eq. 1), h^ =1 k,h? — l,fc^ =. 42) where the functions ex, mx, px, which will be called in this chapter polar cosexponential functions, are defined by the series ex y = 1 + t//3\ + / / 6 ! 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.

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