Euler in his paper De la controverse entre Mrs. Leibniz and Bernoulli sur les logarithmes des nombres négatifs and imaginairesg (Mémoires de l'Académie des Sciences de Berlin 5 (1749), 139-171, in: Opera, (1) 17, 195-232; cf. C. G. Fraser [1]) considered the rule d(log x) = dx/x. He rejected an earlier suggestion of Leibniz that this rule is only valid for positive real values of x with the following observation:"(...) Car, comme ce calcul roule sur les quantités variables, c. à d. sur des quantités...

The term "Algebraic Analysis" in the last two decades is used in two completely different senses. It seems that at least one is far away from its historical roots. Thus, in order to explain this misunderstanding, the history of this term from its origins is recalled.

It is well known that a power of a right invertible operator is again right invertible, as well as a polynomial in a right invertible operator under appropriate assumptions. However, a linear combination of right invertible operators (in particular, their sum and/or difference) in general is not right invertible. It will be shown how to solve equations with linear combinations of right invertible operators in commutative algebras using properties of logarithmic and antilogarithmic mappings. The...

There are applied power mappings in algebras with logarithms induced
by a given linear operator D in order to study particular properties of powers
of logarithms. Main results of this paper will be concerned with the case
when an algebra under consideration is commutative and has a unit and
the operator D satisfies the Leibniz condition, i.e. D(xy) = xDy + yDx for
x, y ∈ dom D. Note that in the Number Theory there are well-known several
formulae expressed by means of some combinations of powers...

CONTENTSPreface...................... 5Acknowledgment...................... 7PART A. LINEAR OPERATORS IN LINEAR SPACESCHAPTER I. Operators with a finite and semifinite dimensional characteristic........ 25CHAPTER II. Algebraic and almost algebraic operators........ 65CHAPTER III. Φ_Ξ-operators........ 90CHAPTER IV. Determinant theory of Φ_Ξ-operators........ 102PART B. LINEAR OPERATORS IN LINEAR TOPOLOGICAL SPACESCHAPTER I. Linear topological and linear metric space........ 115CHAPTER II. Continuous...

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