A novel operation associated with Gauss' arithmetic-geometric means
Shinji Tanimoto
TL;DR
This paper introduces a new binary operation linked to Gauss' AGM, positioning it as an intermediate between addition and multiplication, and explores its algebraic properties.
Contribution
It defines a novel operation associated with AGM and proves several of its algebraic properties, expanding the understanding of means and operations.
Findings
The new operation is mathematically well-defined.
Several algebraic properties of the operation are established.
The operation bridges the conceptual gap between addition and multiplication.
Abstract
The arithmetic mean is the mean for addition and the geometric mean is that for multiplication. Then what kind of binary operation is associated with the arithmetic-geometric mean (AGM) due to C. F. Gauss? If it is possible to construct an arithmetic operation such that AGM is the mean for this operation, it can be regarded as an intermediate operation between addition and multiplication in view of the meaning of AGM. In this paper such an operation is introduced and several of its algebraic properties are proved.
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Taxonomy
TopicsMathematical Inequalities and Applications · Iterative Methods for Nonlinear Equations · Functional Equations Stability Results