Logarithms and Antilogarithms: An Algebraic Analysis Approach / Edition 1 available in Hardcover
This volume proposes and explores a new definition of logarithmic mappings as invertible selectors of multifunctions induced by linear operators with domains and ranges in an algebra over a field of characteristic zero. Several important previously published results are presented. Amongst the applications of logarithmic and antilogarithmic mappings are the solution of linear and nonlinear equations in algebras of square matrices. Some results may also provide numerical algorithms for the approximation of solutions.
Audience: Research mathematicians and other scientists of other disciplines whose work involves the solution of equations.
Table of ContentsPreface. 1. Preliminaries, Introduction to Algebraic Analysis. 2. Basic Equation. Logarithms and Antilogarithms. 3. Logarithms and Antilogarithms of Higher Order. 4. Logarithms and Antilogarithms of Operators Having Either Finite Nullity or Finite Deficiency. 5. Reduction Theorems. 6. Multiplicative Case. 7. Leibniz Algebras. 8. Linear Equations in Leibniz Algebras. 9. Trigonometric Mappings and Elements. 10. Semigroup Properties of Solutions to Linear Equations in Leibniz Algebras. 11. Operator ehD. 12. Power Mappings. Polylogarithmic Functions. Nonlinear Equations. 13. Smooth Logarithms and Antilogarithms. 14. Riemann-Hilbert Problems Type in Leibniz Algebras. 15. Periodic Problems. Generalized Floquet Theorem. 16. Equations with Multiplicative Involutions of Order N. 17. Remarks on the Fractional Calculus. Appendix: Functional Shifts; Z. Binderman. A1. Functions of a Right Invertible Operator. A2. Functional Shifts. A3. Isomorphisms of Spaces of Functional Shifts. A4. Functional Shifts Induced by Operators of Complex Differentiation. A5. Euler-Maclaurin Type Formulae. A6. Differential and Integral Properties. References. Subject Index. Authors Index. List of Symbols.