The Complete List of Solutions to M\"{o}bius's Exponential Equation
DOI:
https://doi.org/10.29020/nybg.ejpam.v18i2.6041Keywords:
Mobius addition, Mobius's Exponential Equation, associator function, gyrogroup, non-linear equationAbstract
M\"{o}bius addition is a non-associative binary operation defined on the complex open unit disk $\mathbb{D} = \{z\in\mathbb{C}:|z|<1\}$ by $a\oplus_M b = \dfrac{a+b}{1+\overline{a}b}$, and M\"{o}bius's Exponential Equation is a non-linear functional equation of the form $L(a\oplus_M b) = L(a)L(b)$, where $L$ is a complex-valued function defined on $\mathbb{D}$. In [Aequat. Math. \textbf{91} (2017), 491--503], the authors address the problem of determining the solutions to M\"{o}bius's Exponential Equation. In this paper, we determine the complete list of solutions to M\"{o}bius's Exponential Equation using an algebraic approach.
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Copyright (c) 2025 Rasimate Maungchang, Watchareepan Atiponrat, Tarid Suwansri, Jaturon Wattanapan, Teerapong Suksumran
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