Cosine and sine addition and subtraction law with an automorphism

Youssef Aserrar; Elhoucien Elqorachi

Published: 2023-11-29

Vol. 38 No. 2 (2024)

Cosine and sine addition and subtraction law with an automorphism

Youssef Aserrar

, Elhoucien Elqorachi

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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Aserrar, Y., & Elqorachi, E. (2023). Cosine and sine addition and subtraction law with an automorphism. Annales Mathematicae Silesianae, 38(2), 155–176. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/16429

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Youssef Aserrar

youssefaserrar05@gmail.com
https://orcid.org/0009-0003-6821-227X

Affiliation:

Department of mathematics, Faculty of sciences, Ibn Zohr University Morocco

Elhoucien Elqorachi

Affiliation:

Department of mathematics, Faculty of sciences, Ibn Zohr University Morocco

Vol. 38 No. 2 (2024)
Published: 2024-07-23

ISSN: 0860-2107

eISSN: 2391-4238

10.1515/amsil

Publisher

University of Silesia Press

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