<abstract xmlns="http://www.w3.org/1999/xhtml">

The primary object of study is the “cosine-sine” functional equation <italic>f</italic>(<italic>xy</italic>) = <italic>f</italic>(<italic>x</italic>)<italic>g</italic>(<italic>y</italic>)+<italic>g</italic>(<italic>x</italic>)<italic>f</italic>(<italic>y</italic>)+<italic>h</italic>(<italic>x</italic>)<italic>h</italic>(<italic>y</italic>) for unknown functions <italic>f, g, h</italic> : <italic>S →</italic> ℂ, where <italic>S</italic> 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 <italic>f</italic>(<italic>xy</italic>) = <italic>f</italic>(<italic>x</italic>)<italic>g</italic>(<italic>y</italic>) + <italic>g</italic>(<italic>x</italic>)<italic>f</italic>(<italic>y</italic>) <italic>− g</italic>(<italic>x</italic>)<italic>g</italic>(<italic>y</italic>) separately, since it has an independent direct solution on a general semigroup.
We give the continuous solutions on topological semigroups for both equations.
</abstract>

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.

The Cosine-Sine Functional Equation on Semigroups

Annales Mathematicae Silesianae

This work is licensed under the Creative Commons Attribution 4.0 International License.

{"article-title":"The Cosine-Sine Functional Equation on Semigroups"}

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...

半群上的余弦函数方程

在线发布:2021年10月5日

页面范围： 30 - 52

收到:2021年5月14日

认可的:2021年8月19日

内政部： https://doi.org/10.2478/amsil-2021-0012

关键词
半群,素理想,正弦加法律,乘法函数,列维-西维塔方程

©2022 Bruce Ebanks，Sciendo出版

本作品根据知识共享署名4.0国际许可证授权。