代数的微分,積分と差分方程式
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概要
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S. Hayabara and S. Haruki published a book [4] "The new operator methods and the theory of discrete analytic functions (in Japanese)". Concerning to the solution of linear sequence equations or difference equations with constant coefficients, the new operator methods are explained in this book. However for sequence equations or difference equations with variable coefficients, it is necessary to express these equations in the opertional space. T. Fenyes and P. Kosik [1] introduced an "Algebraic derivative" such as D{X_n}={-nx_n} and transformed equations of sequences in E-space into algebraic differential equations in Q-space by using operation D. The solutions of algebraic differential equations of the first order was shown in [1] and [4]. In this note we will introduce a new algebraic logarithmic function and obtain several formulae of algebraic integrals. And we will express solutions for ordinary algebraic dfferential equations of the second order and for algebraic partial differential equations.