Abstract
The sine and cosine addition laws on a (not necessarily commutative) semigroup are $$f(xy) = f(x)g(y) + g(x)f(y)$$ , respectively $$g(xy) = g(x)g(y) - f(x)f(y)$$ . Both of these have been solved on groups, and the first one has been solved on semigroups generated by their squares. Quite a few variants and extensions with more unknown functions and/or additional terms have also been studied. Here we extend these results and solve the Levi–Civita functional equation $$f(xy) = g_1(x)h_1(y) + g_2(x)h_2(y)$$ by elementary methods on groups and monoids generated by their squares, assuming that f is central. We also find the continuous solutions in the case of topological groups and monoids.
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