## Continued Fraction Theory

##### Abstract

I have divided this paper on the theory of continued actions into
five sections. The first section consists of general basic theorems in
the theory, many of which - in particular, theorem 1.1 - are used throughout
the remainder of the paper. These include theorems on general formulae,
convergence, remainders, and transformations, the latter being applied to
determine the continued fraction expansion of a power ser1es. Section
two is devoted to simple continued fractions, and in its four divisions
are discussed finite continued fractions, infinite continued fractions,
periodic continued fractions (including the famous theorem of Lagrange on
quadratic irrationals), and last, the approximation of irrationals using
their simple continued fraction expansions. The third section deals with analytic convergence theory, including a general discussion of the
theory, followed by many specific convergence tests. Concepts from the
theory of complex variables are used throughout the section. In the
fourth section, the solution of a general Riccatti differential equation
is developed into a continued fraction expansion, and this is applied to
determine the continued fraction expansions for many familiar functions.
To conclude the section, a discussion of the expansion for the not-so-familiar
hypergeometric. function is presented. The fifth section ends
the paper with a discussion of proper continued fractions, followed by a
treatment of the expansions of quadratic-irrational type functions.