Analytic Continued Fractions

3.5.2 Analytic Continued Fractions

Continued fractions have also important applications in analysis. A large number of analytic functions are known to have continued fraction representations. Indeed, some of the best algorithms for the numerical computation of important analytic functions are based on continued fractions. We shall not give complete proofs but refer to classical books of Perron [289], Wall [369], and Henrici [196, 197].

A continued fraction is said to be equivalent to a given series if and only if the sequence of convergents is equal to the sequence of partial sums . There is typically an infinite number of such equivalent fractions. The construction of the continued fraction is particularly simple if we require that the denominators q n = 1 for all n ≥ 1. For a power series we shall thus have

n z , n ≥ 1. We must assume that c j

n =c 0 +c 1 z

+c n 2 z +···+c

3.5. Continued Fractions and Padé Approximants 327 We shall determine the elements a n ,b n by means of the recursion formulas of Theo-

rem 3.5.1 (for n ≥ 2) with initial conditions. We thus obtain the following equations: p n =b n p n −1 +a n p n −2 , p 0 =b 0 , p 1 =b 0 b 1 +a 1 ,

1=b n +a n ,

b 1 = 1.

The solution reads b 0 =p 0 =c 0 ,b 1 = 1, a 1 =p 1 −p 0 =c 1 z , and for n ≥ 2,

a n = (p n −p n −1 )/(p n −2 −p n −1 ) = −zc n /c n −1 ,

b n =1−a n = 1 + zc n /c n −1 ,

+c −1 1 z +···+c

zc 1 zc 2 /c 1 zc n /c n

1 + zc n /c n −1 − Of course, an equivalent continued fraction gives by itself no convergence accelera-

c 0 n z ···=c 0 +

tion, just because it is equivalent . We shall therefore leave the subject of continued fractions equivalent to a series, after showing two instances of the numerous pretty formulas that can

be obtained by this construction. For

f (z) =e =1+z+

arctan √z

=1− 3+ 5− 7+··· we obtain for z = −1 and z = 1, respectively, after simple equivalence transformations,

1+ ..., 1+y 2+y 2+ 3+ 4+ 5+

There exist, however, other methods to make a correspondence between a power series and a continued fraction. Some of them lead to a considerable convergence acceleration that often makes continued fractions very efficient for the numerical computation of functions. We shall return to such methods in Sec. 3.5.3.

Gauss developed a continued fraction for the ratio of two hypergeometric functions (see (3.1.16)),

where (a

(b + n)(c − a + n) 2n+1 =

. (3.5.16) (c

a + n)(c − b + n)

a 2n

+ 2n)(c + 2n + 1) (c + 2n − 1)(c + 2n) Although the power series converge only in the disk |z| < 1, the continued fraction of Gauss

converges throughout the complex z-plane cut along the real axis from 1 to +∞. It provides an analytic continuation in the cut plane.

328 Chapter 3. Series, Operators, and Continued Fractions If we set b = 0 in (3.5.15), we obtain a continued fraction for F (a, 1, c + 1; z). From

this, many continued fractions for elementary functions can be derived, for example,

arctan z =

The expansion for tan z is valid everywhere, except in the poles. For arctan z the continued fraction represents a single-valued branch of the analytic function in a plane with cuts along the imaginary axis extending from +i to +i∞ and from −i to −i∞. A continued fraction expansion for arctanhz is obtained by using the relation arctanhz = −i arctan iz. In all these cases the region of convergence as well as the speed of convergence is considerably larger than for the power series expansions. For example, the sixth convergent for tan π/4 is almost correct to 11 decimal places.

For the natural logarithm we have

log(1 + z) = ···, (3.5.19)

The fraction for the logarithm can be used in the whole complex plane except for the cuts ( −∞, −1] and [1, ∞). The convergence is slow when z is near a cut. For elementary

functions such as these, properties of the functions can be used for moving z to a domain where the continued fraction converges rapidly.

Example 3.5.3.

Consider the continued fraction for ln(1 + z) and set z = 1. The successive approxi- mations to ln 2 = 0.69314 71806 are the following.

0.693152 Note that the fractions give alternatively upper and lower bounds for ln 2. It can be shown

that this is the case when the elements of the continued fraction are positive. To get the accuracy of the last approximation above would require as many as 50,000 terms of the series ln 2 = ln(1 + 1) = 1 − 1/2 + 1/3 − 1/4 + · · ·.

Continued fraction expansions for the gamma function and the incomplete gamma function are found in the Handbook [1, Sec. 6.5]. For the sake of simplicity we assume that x>

0, although the formulas can be used also in an appropriately cut complex plane. The

3.5. Continued Fractions and Padé Approximants 329 parameter a may be complex in Ŵ(a, x). 118

Ŵ(a, x)

t = a −1 e −t dt, Ŵ(a,

0) = Ŵ(a),

xx

γ (a, x) = Ŵ(a) − Ŵ(a, x) = a t −1 e −t dt, ℜa > 0,

a 1−a

1 2−a 2

Ŵ(a, x) =e −x

=e −x x Ŵ(a)

We mention these functions because they have many applications. Several other important functions can, by simple transformations, be brought to particular cases of this function, for example, the normal probability function, the chi-square probability function, the exponential integral, and the Poisson distribution.

The convergence behavior of continued fraction expansions is much more complicated than for power series. Gautschi [142] exhibits a phenomenon of apparent convergence to the wrong limit for a continued fraction of Perron for ratios of Kummer functions. The sequence of terms initially decreases rapidly, then increases, and finally again decreases to zero at a supergeometric rate.

Continued fractions such as these can often be derived by a theorem of Stieltjes which relates continued fractions to orthogonal polynomials that satisfy a recurrence relation of the same type as the one given above. Another method of derivation is the Padé approximation, studied in the next section, that yields a rational function. Both techniques can be looked upon as a convergence acceleration of an expansion into powers of z or z −1 .