Generalized Continued Fraction

In complex analysis, a branch of mathematics, a generalized continued fraction is a generalization of regular continued fractions in canonical form, in which the partial numerators and partial denominators can assume arbitrary real or complex values.

A generalized continued fraction is an expression of the form

where the an (n > 0) are the partial numerators, the bn are the partial denominators, and the leading term b0 is called the integer part of the continued fraction.

The successive convergents of the continued fraction are formed by applying the fundamental recurrence formulas:


x_0 = \frac{A_0}{B_0} = b_0, \qquad
x_1 = \frac{A_1}{B_1} = \frac{b_1b_0+a_1}{b_1},\qquad
x_2 = \frac{A_2}{B_2} = \frac{b_2(b_1b_0+a_1) + a_2b_0}{b_2b_1 + a_2},\qquad\cdots\,

and in general


A_n = b_n A_{n-1} + a_n A_{n-2}, \qquad
B_n = b_n B_{n-1} + a_n B_{n-2}, \,

where An is the numerator and Bn is the denominator, called continuants, of the nth convergent.

If the sequence of convergents {xn} approaches a limit the continued fraction is convergent and has a definite value. If the sequence of convergents never approaches a limit the continued fraction is divergent. It may diverge by oscillation (for example, the odd and even convergents may approach two different limits), or it may produce an infinite number of zero denominators Bn.

Read more about Generalized Continued Fraction:  History of Continued Fractions, Notation, Some Elementary Considerations, Linear Fractional Transformations, Continued Fractions and Series, Higher Dimensions

Famous quotes containing the words generalized, continued and/or fraction:

    One is conscious of no brave and noble earnestness in it, of no generalized passion for intellectual and spiritual adventure, of no organized determination to think things out. What is there is a highly self-conscious and insipid correctness, a bloodless respectability submergence of matter in manner—in brief, what is there is the feeble, uninspiring quality of German painting and English music.
    —H.L. (Henry Lewis)

    The Sound of battle fell upon my ear & heart all day yesterday—even after dark the cannon’s insatiate roar continued ...
    Elizabeth Blair Lee (1818–?)

    The mother as a social servant instead of a home servant will not lack in true mother duty.... From her work, loved and honored though it is, she will return to her home life, the child life, with an eager, ceaseless pleasure, cleansed of all the fret and fraction and weariness that so mar it now.
    Charlotte Perkins Gilman (1860–1935)