# The Stability of Functional Equations for Trigonometric and Similar Functions

• Donald H. Hyers
• George Isac
• Themistocles M. Rassias
Chapter
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 34)

## Abstract

In this section, we will be interested primarily in the stability of the following equations:

(6.1) $$f\left( {x + y} \right) + f\left( {x - y} \right) = 2f\left( x \right)f\left( y \right)$$ (d’Alembert)

(6.2) $$f{\left( {\frac{{x + y}}{2}} \right)^2} = f\left( x \right)f\left( y \right)$$ (Lobačevskiĭ)

(6.3a) $$f\left( {x + y} \right)f\left( {x - y} \right) = f{\left( x \right)^2} - {\left( y \right)^2}$$ (sine equation)

(6.3b) $$f\left( {x + y} \right) = f\left( x \right)g\left( y \right) + g\left( x \right)f\left( y \right)$$ (sine equation)

(6.4) $$f\left( {x + y} \right) = f\left( x \right)f\left( y \right) - g\left( x \right)g\left( y \right)$$ (cosine equation)

where f and g may be defined on a group or semigroup with values in a field K which usually is the field of real or complex numbers. Methods of solving such equations are described in the books by Aczél (1966) and Aczél and Dhombres (1989).

## Keywords

Abelian Group Functional Equation Trigonometric Function Vector Space Versus Normed Algebra
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Authors and Affiliations

• Donald H. Hyers
• George Isac
• 1
• Themistocles M. Rassias
• 2