Logarithmic and antilogarithmic mappings

Danuta Przeworska-Rolewicz

  • Publisher: Instytut Matematyczny Polskiej Akademi Nauk(Warszawa), 1994

Abstract

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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 considérées en général, s'il n'était pas vrai généralement qu'il fût d· lx = dx/x, quelque quantité qu'on donne à x, soit positive ou négative, ou même imaginaire, on ne pourrait jamais se servir de cette règle, la vérité du calcul différentiel étant fondée sur la généralité des règles qu'il renferme."CONTENTSIntroduction............................................................................................................................................50. Preliminaries.......................................................................................................................................61. Basic equation. Logarithms and antilogarithms..................................................................................82. Logarithms and antilogarithms of higher order.................................................................................193. Reduction theorems..........................................................................................................................244. Multiplicative case.............................................................................................................................365. Leibniz case.......................................................................................................................................416. Exponential, power and polylogarithmic functions.............................................................................517. Complex case....................................................................................................................................578. Smooth logarithms and antilogarithms..............................................................................................649. Logarithmic and antilogarithmic mappings induced by left invertible and invertible operators...........7010. Other generalizations.......................................................................................................................82   References...........................................................................................................861991 Mathematics Subject Classification: 47C05, 47H17, 47S10, 33B10.

How to cite

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Danuta Przeworska-Rolewicz. Logarithmic and antilogarithmic mappings. Warszawa: Instytut Matematyczny Polskiej Akademi Nauk, 1994. <http://eudml.org/doc/268510>.

@book{DanutaPrzeworska1994,
abstract = {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 considérées en général, s'il n'était pas vrai généralement qu'il fût d· lx = dx/x, quelque quantité qu'on donne à x, soit positive ou négative, ou même imaginaire, on ne pourrait jamais se servir de cette règle, la vérité du calcul différentiel étant fondée sur la généralité des règles qu'il renferme."CONTENTSIntroduction............................................................................................................................................50. Preliminaries.......................................................................................................................................61. Basic equation. Logarithms and antilogarithms..................................................................................82. Logarithms and antilogarithms of higher order.................................................................................193. Reduction theorems..........................................................................................................................244. Multiplicative case.............................................................................................................................365. Leibniz case.......................................................................................................................................416. Exponential, power and polylogarithmic functions.............................................................................517. Complex case....................................................................................................................................578. Smooth logarithms and antilogarithms..............................................................................................649. Logarithmic and antilogarithmic mappings induced by left invertible and invertible operators...........7010. Other generalizations.......................................................................................................................82   References...........................................................................................................861991 Mathematics Subject Classification: 47C05, 47H17, 47S10, 33B10.},
author = {Danuta Przeworska-Rolewicz},
keywords = {operational calculus; logarithmic and antilogarithmic mappings; -algebra; linear equations; algebras with left invertible and invertible operators; logarithms; antilogarithms; finite nullity; finite deficiency},
language = {eng},
location = {Warszawa},
publisher = {Instytut Matematyczny Polskiej Akademi Nauk},
title = {Logarithmic and antilogarithmic mappings},
url = {http://eudml.org/doc/268510},
year = {1994},
}

TY - BOOK
AU - Danuta Przeworska-Rolewicz
TI - Logarithmic and antilogarithmic mappings
PY - 1994
CY - Warszawa
PB - Instytut Matematyczny Polskiej Akademi Nauk
AB - 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 considérées en général, s'il n'était pas vrai généralement qu'il fût d· lx = dx/x, quelque quantité qu'on donne à x, soit positive ou négative, ou même imaginaire, on ne pourrait jamais se servir de cette règle, la vérité du calcul différentiel étant fondée sur la généralité des règles qu'il renferme."CONTENTSIntroduction............................................................................................................................................50. Preliminaries.......................................................................................................................................61. Basic equation. Logarithms and antilogarithms..................................................................................82. Logarithms and antilogarithms of higher order.................................................................................193. Reduction theorems..........................................................................................................................244. Multiplicative case.............................................................................................................................365. Leibniz case.......................................................................................................................................416. Exponential, power and polylogarithmic functions.............................................................................517. Complex case....................................................................................................................................578. Smooth logarithms and antilogarithms..............................................................................................649. Logarithmic and antilogarithmic mappings induced by left invertible and invertible operators...........7010. Other generalizations.......................................................................................................................82   References...........................................................................................................861991 Mathematics Subject Classification: 47C05, 47H17, 47S10, 33B10.
LA - eng
KW - operational calculus; logarithmic and antilogarithmic mappings; -algebra; linear equations; algebras with left invertible and invertible operators; logarithms; antilogarithms; finite nullity; finite deficiency
UR - http://eudml.org/doc/268510
ER -

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