Addition formulae with singularities


We deal with functional equations of the form
f(x+y) = F(f(x),f(y))
(so called addition formulas) assuming that the given binary operation F is associative but its domain of definition is disconnected (admits "singularities"). The function
Flu,v) := (u+v)/(1+uv)
serves here as a good example; the corresponding equation characterizes the hyperbolic tangent. Our considerations may be viewed as counterparts of L. Losonczi's [4] and K. Domańska's [2] results on local solutions of the functional equation
f(F(x,y)) = f(x) + f(y)
with the same behaviour of the given associative operation F.
Our results exhibit a crucial role of 1 that turns out to be the critical value towards the range of the unknown function. What concerns the domain we admit fairly general structures (groupoids, groups, 2-divisible groups). In the case where the domain forms a group admitting subgroups of index 2 the family of solutions enlarges considerably.

1. K. Dankiewicz, Z. Moszner, Prolongements des homomorphismes et des solutions de l'equation de translation, Annales de l'École Normale Supérieure à Cracovie, Travaux Math. 10 (1982), 27-44.
2. K. Domańska, Cauchy type equations related to some singular associative operations, Glasnik Mat. 31 (1996), 135-149.
3. M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, Polish Scientific Publishers & Silesian University, Warszawa-Kraków-Katowice 1985.
4. L. Losonczi, Local solutions of functional equations, Glasnik Mat. 25 (1990), 57-67.

Published : 2004-09-30

DomańskaK., & GerR. (2004). Addition formulae with singularities. Annales Mathematicae Silesianae, 18, 7-20. Retrieved from

Katarzyna Domańska
Instytut Matematyki i Informatyki, Akademia im. Jana Długoosza w Częstochowie  Poland
Roman Ger 
Instytut Matematyki, Uniwersytet Śląski w Katowicach  Poland

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