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Continued Fractions: Analytic Theory and Applications by William B. Jones, W. J. Thron

By William B. Jones, W. J. Thron

This can be an exposition of the analytic concept of persisted fractions within the complicated area with emphasis on functions and computational tools.

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Extra resources for Continued Fractions: Analytic Theory and Applications (Encyclopedia of Mathematics and its Applications (No. 11))

Example text

Hereafter when we speak of a continued fraction over a normed field F, other than C, we shall always specify the field F. If no field is specified, it is understood that we are in the field of complex numbers. Our main application of continued fractions over general normed fields comes in Chapter 5 when we consider the field £ of formal Laurent series. 2, we first consider some elementary examples of continued fractions and some methods for computing approximants of continued fractions. 15b) where x _x.

_ x. ,/i. In Chapter 10 it is shown that in many circumstances the BR algorithm is numerically stable in the sense that the rounding error produced in the computation of fn either is bounded or else grows very slowly as n increases. This suggests that the BR algorithm may be particularly suitable for computational purposes. What makes the BR algorithm less attractive than other (forward) algorithms is the uncertainty as to the proper value of n. TS' "X-V^O, ,1-0,1,2,.... t. is sometimes called a Moebius transformation.

1)). Then since sn(0)^sn(co) and hence S^^s^^S^^s^oo)), are non-singular. Since fo = bo it must be in C. 7 is that it is possible to have a continued fraction which converges at any speed (fast or slow) or which diverges in any manner. 8. A sequence {/„} in the extended complex plane can be the sequence of approximants of a continued fraction of the form bo-\-¥j 0, let An=fn and Bn=\ if fn is finite, and let An=\ and Bn = 0 if fn=oo.

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