Ebook: Logarithms and Antilogarithms: An Algebraic Analysis Approach

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.

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Content:
Front Matter....Pages i-ix
Preliminaries. Introduction to Algebraic Analysis....Pages 1-10
Basic Equation. Logarithms and Antilograrithms....Pages 11-38
Logarithms and Antilogarithms of Higher Order....Pages 39-51
Logarithms and Antilogarithms of Operators Having Either Finite Nullity or Finite Deficiency....Pages 52-62
Reduction Theorems....Pages 63-81
Multiplicative Case....Pages 82-91
Leibniz Algebras....Pages 92-112
Linear Equations in Leibniz Algebras....Pages 113-135
Trigonometric Mappings and Elements....Pages 136-148
Semigroups Properties of Solutions to Linear Equations....Pages 149-155
Operator ehD ....Pages 156-166
Power Mappings. Polylogarithmic Functions. Nonlinear Equations....Pages 167-186
Smooth Logarithms and Antilogarithms....Pages 187-200
Reimann-Hilbert Type Problems in Leibniz Algebras....Pages 201-215
Periodic Problems....Pages 216-235
Equations with Multiplicative Involutions of Order N ....Pages 236-255
Remarks on the Fractional Calculus....Pages 256-258
Back Matter....Pages 259-357

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Content:
Front Matter....Pages i-ix
Preliminaries. Introduction to Algebraic Analysis....Pages 1-10
Basic Equation. Logarithms and Antilograrithms....Pages 11-38
Logarithms and Antilogarithms of Higher Order....Pages 39-51
Logarithms and Antilogarithms of Operators Having Either Finite Nullity or Finite Deficiency....Pages 52-62
Reduction Theorems....Pages 63-81
Multiplicative Case....Pages 82-91
Leibniz Algebras....Pages 92-112
Linear Equations in Leibniz Algebras....Pages 113-135
Trigonometric Mappings and Elements....Pages 136-148
Semigroups Properties of Solutions to Linear Equations....Pages 149-155
Operator ehD ....Pages 156-166
Power Mappings. Polylogarithmic Functions. Nonlinear Equations....Pages 167-186
Smooth Logarithms and Antilogarithms....Pages 187-200
Reimann-Hilbert Type Problems in Leibniz Algebras....Pages 201-215
Periodic Problems....Pages 216-235
Equations with Multiplicative Involutions of Order N ....Pages 236-255
Remarks on the Fractional Calculus....Pages 256-258
Back Matter....Pages 259-357
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