The notation used throughout the DLMF for the continued fraction
is called the th approximant or convergent to . and are called the th (canonical) numerator and denominator respectively.
Two continued fractions are equivalent if they have the same convergents.
there is a sequence , ,
, such that
when , .
when , .
A sequence in the extended complex plane, , can be a sequence of convergents of the continued fraction (1.12.3) iff
A contraction of a continued fraction is a continued fraction whose convergents form a subsequence of the convergents of . Conversely, is called an extension of . If , , then is called the even part of . The even part of exists iff , , and up to equivalence is given by
If , , then is called the odd part of . The odd part of exists iff , , and up to equivalence is given by
A continued fraction converges if the convergents tend to a finite limit as .
The continued fraction converges when
With these conditions the convergents satisfy and with .
Let the elements of the continued fraction satisfy
where is an arbitrary small positive constant. Then the convergents satisfy
and the even and odd parts of the continued fraction converge to finite values. The continued fraction converges iff, in addition,
In this case .
For analytical and numerical applications of continued fractions to special functions see §3.10.