#### Date on Master's Thesis/Doctoral Dissertation

8-2014

#### Document Type

Doctoral Dissertation

#### Degree Name

Ph. D.

#### Department

Mathematics

#### Committee Chair

Sahoo, Prasanna K.

#### Committee Member

Brown, David

#### Committee Member

Gill, Ryan

#### Committee Member

Powers, Robert

#### Committee Member

Riedel, Thomas

#### Subject

Functional equations

#### Abstract

Let G be an abelian group, C be the _eld of complex numbers, _ 2 G be any _xed, nonzero element and _ : G ! G be an involution. In Chapter 2, we determine the general solution f; g : G ! C of the functional equation f(x + _y + _) + g(x + y + _) = 2f(x)f(y) for all x; y 2 G. Let G be an arbitrary group, z0 be any _xed, nonzero element in the center Z(G) of the group G, and _ : G ! G be an involution. The main goals of Chapter 3 are to study the functional equations f(x_yz0) ?? f(xyz0) = 2f(x)f(y) and f(x_yz0) + f(xyz0) = 2f(x)f(y) for all x; y 2 G and some _xed element z0 in the center Z(G) of the group G. In Chapter 4, we consider some properties of the general solution to f(xy)f(x_y) = f(x)2 ?? f(y)2. We also _nd the solution to this equation when G is a 2-divisible, perfect group. We end the chapter by discussing the periodicity of the solutions to both the sine functional equation and the sine inequality.

#### Recommended Citation

Perkins, Allison, "Functional equations with involution related to sine and cosine functions." (2014). *Electronic Theses and Dissertations.* Paper 1117.

https://doi.org/http://dx.doi.org/10.18297/etd/1117