The cosine-sine functional equation on semigroups

Bruce Ebanks


The primary object of study is the “cosine-sine” functional equation f(xy) = f(x)g(y)+g(x)f(y)+h(x)h(y) for unknown functions f,g,h:S→ℂ, where S is a semigroup. The name refers to the fact that it contains both the sine and cosine addition laws. This equation has been solved on groups and on semigroups generated by their squares. Here we find the solutions on a larger class of semigroups and discuss the obstacles to finding a general solution for all semigroups. Examples are given to illustrate both the results and the obstacles.
We also discuss the special case f(xy) = f(x)g(y)+g(x)f(y)–g(x)g(y) separately, since it has an independent direct solution on a general semigroup.
We give the continuous solutions on topological semigroups for both equations.


semigroup; prime ideal; sine addition law; multiplicative function; Levi-Civita equation

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Published : 2021-10-05

EbanksB. (2021). The cosine-sine functional equation on semigroups. Annales Mathematicae Silesianae, 36(1), 30-52. Retrieved from

Bruce Ebanks
Department of Mathematics, University of Louisville, USA  United States

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