, for all , Even and odd convergents It is sometimes necessary to separate a continued fraction into its even and odd parts. For example, if the continued fraction diverges by oscillation between two distinct limit points and , then the sequence must converge to one of these, and must converge to the other. In such a situation it may be convenient to express the original continued fraction as two different continued fractions, one of them converging to , and the other converging to . The formulas for the even and odd parts of a continued fraction can be written most compactly if the fraction has already been transformed so that all its partial denominators are unity. Specifically, if is a continued fraction, then the even part and the odd part are given by and respectively. More precisely, if the successive convergents of the continued fraction are , then the successive convergents of as written above are , and the successive convergents of are . Conditions for irrationality If and are positive integers with for all sufficiently large , then converges to an irrational limit. Fundamental recurrence formulas The partial numerators and denominators of the fraction's successive convergents are related by the fundamental recurrence formulas: The continued fraction's successive convergents are then given by These recurrence relations are due to John Wallis (1616–1703) and Leonhard Euler (1707–1783). As an example, consider the regular continued fraction in canonical form that represents the golden ratio : Applying the fundamental recurrence formulas we find that the successive numerators are and the successive denominators are , the Fibonacci numbers. Since all the partial numerators in this example are equal to one, the determinant formula assures us that the absolute value of the difference between successive convergents approaches zero quite rapidly. Linear fractional transformations A linear fractional transformation (LFT) is a complex function of the form where is a complex variable, and are arbitrary complex constants such that . An additional restriction that is customarily imposed, to rule out the cases in which is a constant. The linear fractional transformation, also known as a Möbius transformation, has many fascinating properties. Four of these are of primary importance in developing the analytic theory of continued fractions. If the LFT has one or two fixed points. This can be seen by considering the equation which is clearly a quadratic equation in . The roots of this equation