Generalized continued fraction

Generalized Continued Fraction

Generalized continued fraction

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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

<math>x = b_0 + cfrac</math>

where the a<sub>n</sub> (n &gt; 0) are the partial numerators, the b<sub>n</sub> are the partial denominators, and the leading term b<sub>0</sub> 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 = b_0, qquadx_1 = frac = frac,qquadx_2 = frac = frac,qquadcdots,</math>

where A<sub>n</sub> is the numerator and B<sub>n</sub> is the denominator, called continuant, of the nth convergent.

If the sequence of convergents 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 B<sub>n</sub>.

History of continued fractions

The story of continued fractions begins with the Euclidean......
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