Functional Equations and their Applications

By: Emil Lavlov '14

Advising Faculty: Warren Johnson

Functional equations trace their roots back to 1747 when Jean-Baptiste le Rond d'Alembert was searching for all functions f; g; and h satisfying the equation

f(x + y) ???? f(x ???? y) = g(x)h(y)

for all real numbers x and y: The literature on this subject has many applications not only in mathematics, but also in other sciences such as statistics, physics, nance, and economics. In this thesis, we aim to introduce the reader to functional equations and the methods that can be used to solve them. The paper considers some of the most famous functional equations, including the four Cauchy equations and the equations of d'Alembert, Jensen, Pompeiu, and Hosszu. We also examine generalized versions of some of these equations and many others, allowing us to present the reader with a wide range of functional equations, including equations with more than one unknown function, and equations where the functions may have more than one argument. The solution techniques for most of the equations rely on plugging in di erent values for the variables and manipulating the results. Sometimes theorems from real analysis or linear algebra are also necessary to deal with the problem at hand. We also illustrate several applications of functional equations to di erent areas of science by demonstrating how we can use our knowledge of such equations to derive the formulas for the area of a rectangle, the survival function of an exponential random variable, the scalar and vector products, the future value of an investment, and the Cobb-Douglas production function. We conclude by presenting original research, where we solve some problems that we believe have not been considered before. In addition, we illustrate some new techniques that can be used to tackle a functional equation and show how di erent methods of solution may be combined to nd the unknown functions.

Related Fields: Mathematics