Preprint Article Version 4 Preserved in Portico This version is not peer-reviewed

# Exponentials and Logarithms Properties in an Extended Complex Number Field

Version 1 : Received: 6 April 2021 / Approved: 7 April 2021 / Online: 7 April 2021 (15:20:39 CEST)
Version 2 : Received: 11 April 2021 / Approved: 12 April 2021 / Online: 12 April 2021 (14:31:00 CEST)
Version 3 : Received: 19 September 2021 / Approved: 20 September 2021 / Online: 20 September 2021 (12:09:28 CEST)
Version 4 : Received: 28 November 2021 / Approved: 29 November 2021 / Online: 29 November 2021 (11:11:55 CET)
Version 5 : Received: 14 September 2022 / Approved: 14 September 2022 / Online: 14 September 2022 (03:53:54 CEST)

How to cite: Tischhauser, D. Exponentials and Logarithms Properties in an Extended Complex Number Field. Preprints 2021, 2021040207. https://doi.org/10.20944/preprints202104.0207.v4 Tischhauser, D. Exponentials and Logarithms Properties in an Extended Complex Number Field. Preprints 2021, 2021040207. https://doi.org/10.20944/preprints202104.0207.v4

## Abstract

It is well established the complex exponential and logarithm are multivalued functions, both failing to maintain most identities originally valid over the positive integers domain. Moreover the general case of complex logarithm, with a complex base, is hardly mentionned in mathematic litterature. We study the exponentiation and logarithm as binary operations where all operands are complex. In a redefined complex number system using an extension of the C field, hereafter named E, we prove both operations always produce single value results and maintain the validity of identities such as logu (w v) = logu (w) + logu (v) where u, v, w in E. There is a cost as some algebraic properties of the addition and subtraction will be diminished, though remaining valid to a certain extent. In order to handle formulas in a C and E dual number system, we introduce the notion of set precision and set truncation. We show complex numbers as defined in C are insufficiently precise to grasp all subtleties of some complex operations, as a result multivaluation, identity failures and, in specific cases, wrong results are obtained when computing exclusively in C. A geometric representation of the new complex number system is proposed, in which the complex plane appears as an orthogonal projection, and where the complex logarithm and exponentiation can be simply represented. Finally we attempt an algebraic formalization of E.

## Keywords

Complex number field; Complex exponentiation; Complex logarithm; Exponential and logarithm identities

## Subject

Computer Science and Mathematics, Algebra and Number Theory

Comment 1
Commenter: Daniel Tischhauser
Commenter's Conflict of Interests: Author
+ Respond to this comment