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Logarithmic and antilogarithmic mappings

Seria
Rozprawy Matematyczne tom/nr w serii: 337 wydano: 1994
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Warianty tytułu
Abstrakty
EN

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."
EN

CONTENTS
Introduction............................................................................................................................................5
0. Preliminaries.......................................................................................................................................6
1. Basic equation. Logarithms and antilogarithms..................................................................................8
2. Logarithms and antilogarithms of higher order.................................................................................19
3. Reduction theorems..........................................................................................................................24
4. Multiplicative case.............................................................................................................................36
5. Leibniz case.......................................................................................................................................41
6. Exponential, power and polylogarithmic functions.............................................................................51
7. Complex case....................................................................................................................................57
8. Smooth logarithms and antilogarithms..............................................................................................64
9. Logarithmic and antilogarithmic mappings induced by left invertible and invertible operators...........70
10. Other generalizations.......................................................................................................................82
   References...........................................................................................................86
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne tom/nr w serii: 337
Liczba stron
87
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CCCXXXVII
Daty
wydano
1994
otrzymano
1993-04-30
poprawiono
1994-04-19
Twórcy
Bibliografia
  • Z. Binderman [1], Functional shifts induced by right invertible operators, Math. Nachr. 157 (1992), 211-224.
  • Z. Binderman [2], A unified approach to shifts induced by right invertible operators, Math. Nachr. 161 (1993), 239-252.
  • J. B. Conway and B. B. Morrel [1], Roots and logarithms of bounded operators on a Hilbert space, J. Funct. Anal. 70 (1987),171-193.
  • Z. Dudek [1], On decompositions of quasi-Leibniz D-R algebras, Demonstratio Math. 14 (1981), 745-757.
  • Z. Dudek [2], On multiplicative operators in commutative algebras, Demonstratio Math. 23 (1990), 921-928.
  • C. G. Fraser [1], The calculus of algebraic analysis: some observations on mathematical analysis in the 18th century, Arch. Hist. Exact Sci. 39 (1988), 317-335.
  • H. Kornacki [1], Non-Leibniz components in non-commutative algebras with unit, Demonstratio Math. 18 (1986), 483-497.
  • M. Kuczma [1], An Introduction to the Theory of Functional Equations and Inequalities. Cauchy's Equation and Jensen's Equation, PWN-Polish Scientific Publishers and the Silesian University, Warszawa-Kraków-Katowice, 1985.
  • H. Lausch and D. Przeworska-Rolewicz [1], Pseudocategories, paraalgebras and linear operators, Math. Nachr. 138 (1988), 67-87.
  • H. Lausch and D. Przeworska-Rolewicz [2], Some functional equations appearing in Algebraic Analysis, Opuscula Math. 6 (1990), 111-122.
  • S. J. Lee and M. Z. Nashed [1], Algebraic and topological selections of multi-valued linear relations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 17 (1990), 111-126.
  • D. E. Loeb and G.-C. Rota [1], Formal power series of logarithmic type, Adv. Math. 75 (1989), 1-118.
  • J. B. Miller [1], The standard summation operator, the Euler-Maclaurin sum formula and the Laplace transformation, J. Austral. Math. Soc. 39 (1985), 376-390.
  • Nguyen Van Mau [1], Generalized algebraic elements and singular integral equations with transformed arguments, Wydawnictwa Politechniki Warszawskiej (Warsaw University of Technology Publications), Warszawa, 1989.
  • Nguyen Van Mau [2], Boundary value problems and controllability of linear systems with right invertible operators, Dissertationes Math. 316 (1992).
  • D. Przeworska-Rolewicz [1], Algebraic Analysis, PWN-Polish Scientific Publishers and D. Reidel, Warszawa-Dordrecht, 1988.
  • D. Przeworska-Rolewicz [2], Spaces of D-paraanalytic elements, Dissertationes Math. 302 (1990).
  • D. Przeworska-Rolewicz [3], Advances of one-dimensional kernels, Math. Nachr. 149 (1990), 133-147.
  • D. Przeworska-Rolewicz [4], Commutators with right invertible operators, J. Math. Anal. Appl. 158 (1991), 414-426.
  • D. Przeworska-Rolewicz [5], True shifts, J. Math. Anal. Appl. 170 (1992), 27-48.
  • D. Przeworska-Rolewicz [6], Generalized Bernoulli operator and Euler-Maclaurin formula, in: Advances of Optimization, Proc. 6-th French-German Conference on Optimization, Lambrecht (FRG), 2-8 June 1991, Lecture Notes in Econom. and Math. Systems 382, Springer, Berlin, 1992, 355-368.
  • D. Przeworska-Rolewicz [7], The operator exp(hD) and its inverse formula, Demonstratio Math. 26 (1993), 545-552.
  • D. Przeworska-Rolewicz [8], D-algebras with logarithms, Math. Nachr. 161 (1993), 321-344.
  • D. Przeworska-Rolewicz [9], On logarithmic and antilogarithmic mappings induced by right invertible operators, 1. Real case, preprint no. 508, Institute of Mathematics, Polish Acad. Sci., Warszawa, 1993.
  • D. Przeworska-Rolewicz and S. Rolewicz [1], Equations in Linear Spaces, PWN-Polish Scientific Publishers, Warszawa, 1968.
  • D. Przeworska-Rolewicz and H. von Trotha [1], Right inverses in D-R algebras with unit, J. Integral Equations 3 (1981), 245-259.
  • G. I. Targonski [1], Seminar on Functional Operators and Equations, Lecture Notes in Math. 33, Springer, Berlin, 1967.
  • G. Virsik [1], Right inverses of vector fields, Analysis Paper 84, Dept. of Math., Monash University, Clayton (Melbourne), November 1992.
  • D. Zagier [1], The Bloch-Wigner-Ramakrishnan polylogarithm function, Math. Ann. 286 (1990), 613-624.
Języki publikacji
EN
Uwagi
1991 Mathematics Subject Classification: 47C05, 47H17, 47S10, 33B10.
Identyfikator YADDA
bwmeta1.element.zamlynska-ba01965e-01ed-4379-901a-51724be64112
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ISSN
0012-3862
Kolekcja
DML-PL
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