Logarithmic hyperseries

Abstract

We define the field L of logarithmic hyperseries, construct on L natural operations of differentiation, integration, and composition, establish the basic properties of these operations, and characterize these operations uniquely by such properties.

Surreal numbers, exponentiation and derivations

“…The search for a satisfactory definition of log α (x) for a general x ∈ No >N is connected to the problem of representing No as a field of "hyperseries". For some recent work on this project see [5,46,7].…”

Electronic Structure of Iron.

“…In this paper, we will strongly rely on previous work from [5,17,7,8,6]. The main results from these previous papers will be recalled in Sections 2, 3, and 4.…”

“…The field of logarithmic hyperseries L was defined and studied in [17]. It is a field of Hahn series L = R[[L]] in the sense of [27] that is equipped with a logarithm log : L > −→ L, a derivation ∂ : L −→ L, and a composition • : L × L >,≻ −→ L. Moreover, for each ordinal α ∈ On, it contains an element ℓ α such that…”

“…The first systematic construction of hyperserial fields of strength α < ω ω is due to Schmeling [38]. In this paper, we will rely on the more recent constructions from [17,7] that are fully general. In particular, the surreal numbers No form a hyperserial field in the sense of [7], when equipped with the hyperexponentials and hyperlogarithms from [6].…”

“…This led the second author to conjecture [32, page 16] that there exists a field H of suitably generalized hyperseries in x such that each surreal number can uniquely be represented as the value f (ω) of a hyperseries f ∈ H at x = ω. In order to prove this conjecture, quite some machinery has been developed since: a systematic theory of surreal substructures [5], sufficiently general notions of hyperserial fields [17,7], and definitions of (E α ) α∈No on the surreals that give No the structure of a hyperserial field [8,6]. Now one characteristic property of generalized hyperseries in H should be that they can uniquely be described using suitable expressions that involve x, real numbers, infinite summation, hyperlogarithms, hyperexponentials, and a way to disambiguate nested expansions.…”