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.

**Read Online or Download Complex Numbers in Dimensions PDF**

**Similar number theory books**

**An Introduction to the Theory of Numbers**

The 5th version of 1 of the normal works on quantity conception, written by way of internationally-recognized mathematicians. Chapters are quite self-contained for better flexibility. New gains comprise multiplied therapy of the binomial theorem, suggestions of numerical calculation and a bit on public key cryptography.

**Reciprocity Laws: From Euler to Eisenstein**

This publication is set the improvement of reciprocity legislation, ranging from conjectures of Euler and discussing the contributions of Legendre, Gauss, Dirichlet, Jacobi, and Eisenstein. Readers a professional in easy algebraic quantity conception and Galois conception will locate distinct discussions of the reciprocity legislation for quadratic, cubic, quartic, sextic and octic residues, rational reciprocity legislation, and Eisenstein's reciprocity legislation.

**Discriminant Equations in Diophantine Number Theory**

Discriminant equations are a huge classification of Diophantine equations with shut ties to algebraic quantity concept, Diophantine approximation and Diophantine geometry. This ebook is the 1st accomplished account of discriminant equations and their purposes. It brings jointly many features, together with potent effects over quantity fields, powerful effects over finitely generated domain names, estimates at the variety of recommendations, purposes to algebraic integers of given discriminant, strength indispensable bases, canonical quantity platforms, root separation of polynomials and aid of hyperelliptic curves.

- Quantization and Non-holomorphic Modular Forms
- Topics in Number Theory
- Alice in Numberland: A Students’ Guide to the Enjoyment of Higher Mathematics
- Intersections of Hirzebruch–Zagier Divisors and CM Cycles

**Additional resources for Complex Numbers in Dimensions**

**Sample text**

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.