TY - JOUR
AU - Katarzyna Domańska
AU - Roman Ger
PY - 2004/09/30
Y2 - 2024/06/22
TI - Addition formulae with singularities
JF - Annales Mathematicae Silesianae
JA - AMSil
VL - 18
IS - 0
SE -
DO -
UR - https://www.journals.us.edu.pl/index.php/AMSIL/article/view/14088
AB - We deal with functional equations of the formf(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 functionFlu,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 equationf(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.
ER -