4.5.4 Continued Fractions for Special Cases
Euler used a method of writing an equivalent continued fraction for Maclaurin series. This fraction has convergents which reduce to truncated Maclaurin series.
A short calculation with the recurrence relations (4.4) reveals that
Needless to say, the convergence of the continued fraction can be no different from that of the original series.
Thron's general ^-fraction [Thron, 1948] is a useful modern development. It is given by
with all ej =^0. This fraction is designed to be able to match the Maclaurin series of a function, and its asymptotic expansion expressed as a power series in z"1. It is often called a two-point Padé approximant. If all the e, = 1 and all the dt >0, the T-fraction has the integral representation
where <j>(t) is bounded and nondecreasing on [0, oo). These T-fractions are further discussed in Part II, Section 1.1. A solution for the rational interpolation problem may always be expressed in the form of a convergent of a continued fraction interpolant. These ideas are explained in Part II, Section 1.1.
Exercise 1. Verify Equation (5.7).
Exercise 2. Use the results of Section 4.3, Exercise 1, to verify that
Expansion (i) is designed to be accurate for small |z|, (ii) for large |z|, and (iii) for all |z|. Show that
converges for all z except on cuts from ± / to ± /'oo,
converges for all z except on a cut from i to —i,
converges for all z except on the semicircle |z| = 1, Rez<0.