Cosine and sine addition and subtraction law with an automorphism
Abstract
Let S be a semigroup. Our main result is that we describe the complex-valued solutions of the following functional equations
g(xσ(y)) = g(x)g(y) + f(x)f(y), x, y ∈ S,
f(xσ(y)) = f(x)g(y) + f(y)g(x), x, y ∈ S,
and
f(xσ(y)) = f(x)g(y) - f(y)g(x), x, y ∈ S,
where σ : S → S is an automorphism that need not be involutive. As a consequence we show that the first two equations are equivalent to their variants.
We also give some applications.
Keywords
functional equation; semigroup; addition law; automorphism
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Department of mathematics, Faculty of sciences, Ibn Zohr University Morocco
https://orcid.org/0009-0003-6821-227X
Department of mathematics, Faculty of sciences, Ibn Zohr University Morocco
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