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By William B. Jones, W. J. Thron

This can be an exposition of the analytic thought of persisted fractions within the complicated area with emphasis on purposes and computational equipment.

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141592654 is the approximate value of IT rounded in the 9 th decimal place. 21), we obtain the expansion 1 e = 1 T-7 1 + 1 1 2-3 + 1 ,~ . 17). 4. 4. 718281828 is the approximate value of e rounded in the 9 th decimal place. (an/bn). 6) to obtain fn=An/Bn. multiplication or division are required to obtain fn. ,An can b e computed with only n — 1 additional divisions. 24a) and compute successively from "tail to head" k = n, / i - l , . . 24b) to obtain ^ = Z>OH-G^/|). Only n operations of multiplication or division are required to compute fn.

A procedure for computing pr(t\ which employs the qd algorithm, is described, and a numerical example is given. Finally in Chapter 12 we give some recent results on T-fractions and general T-fractions (mainly due to H. Waadeland) as well as brief accounts of theorems on the location of singular points of analytic functions and on univalency of functions represented by continued fractions. 1 Preliminaries Basic Definitions and Theorems A continued fraction is an ordered pair «{«„},{*„}>,{/„}>, where av a2,...

7) and similar formulas hold for the corresponding "star" sequences. 3 Suppose that the two continued fractions are equivalent. 3) is by induction. If fn and f* denote the nth approximants of bo + K(aH/bH) and 6J + K « / A * ) , respectively, then Sn(0)=fn=fn* = S:(0), n-0,1,2,.... 8) Thus so that S0(W) = SQ(W) and hence Since Sf(O) = Sl(O), it is easy to see that a\/b\ = ax/bx and hence that there exists an r ^ O such that af = rorlal and b1[ = rlbl. 9m. 7). Hence there exists a non-zero constant r m+1 such that a* =r r a ^ ^* =r b .

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