<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...

The Cosine-Sine Functional Equation on Semigroups

Published Online: Oct 05, 2021

Page range: 30 - 52

Received: May 14, 2021

Accepted: Aug 19, 2021

DOI: https://doi.org/10.2478/amsil-2021-0012

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

© 2022 Bruce Ebanks, published by Sciendo

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

The Cosine-Sine Functional Equation on Semigroups

Published Online: Oct 05, 2021

Page range: 30 - 52

Received: May 14, 2021

Accepted: Aug 19, 2021

DOI: https://doi.org/10.2478/amsil-2021-0012

Keywordssemigroup, prime ideal, sine addition law, multiplicative function, Levi-Civita equation

© 2022 Bruce Ebanks, published by Sciendo

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

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