the study of Diophantine equations. In the late eighteenth century Lagrange used continued fractions to construct the general solution of Pell's equation, thus answering a question that had fascinated mathematicians for more than a thousand years. Amazingly, Lagrange's discovery implies that the canonical continued fraction expansion of the square root of every non-square integer is periodic and that, if the period is of length , it contains a palindromic string of length . In 1813 Gauss derived from complex-valued hypergeometric functions what is now called Gauss's continued fractions. They can be used to express many elementary functions and some more advanced functions (such as the Bessel functions), as continued fractions that are rapidly convergent almost everywhere in the complex plane. Notation The long continued fraction expression displayed in the introduction is probably the most intuitive form for the reader. Unfortunately, it takes up a lot of space in a book (and is not easy for the typesetter, either). So mathematicians have devised several alternative notations. One convenient way to express a generalized continued fraction looks like this: Pringsheim wrote a generalized continued fraction this way: Carl Friedrich Gauss evoked the more familiar infinite product when he devised this notation: Here the "" stands for Kettenbruch, the German word for "continued fraction". This is probably the most compact and convenient way to express continued fractions; however, it is not widely used by English typesetters. Some elementary considerations Here are some elementary results that are of fundamental importance in the further development of the analytic theory of continued fractions. Partial numerators and denominators If one of the partial numerators is zero, the infinite continued fraction is really just a finite continued fraction with fractional terms, and therefore a rational function of to and to . Such an object is of little interest from the point of view adopted in mathematical analysis, so it is usually assumed that all . There is no need to place this restriction on the partial denominators . The determinant formula When the th convergent of a continued fraction is expressed as a simple fraction we can use the determinant formula to relate the numerators and denominators of successive convergents and to one another. The proof for this can be easily seen by induction. Base case The case results from a very simple computation. Inductive step Assume that () holds for . Then we need to see the