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.- 11. Operator ehD.- 12. Power mappings. Polylogarithmic functions. Nonlinear equations.- 13. Smooth logarithms and antilogarithms.- 14. Riemann-Hilbert type problems in Leibniz algebras.- 15. Periodic problems.- 16. Equations with multiplicative involutions of order N.- 17. Remarks on the fractional calculus.- Appendix. Functional shifts. By 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.- Authors Index.- List of Symbols.