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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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On the convergence behavior of continued fractions with real elements
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by Walter Gautschi PDF
Math. Comp. 40 (1983), 337-342 Request permission


We define the notion of transient (geometric) convergence rate for infinite series and continued fractions. For a class of continued fractions with real elements we prove a monotonicity property for such convergence rates which helps explain the effectiveness of certain continued fractions known to converge "only" sublinearly. This is illustrated in the case of Legendre’s continued fraction for the incomplete gamma function.
  • Walter Gautschi, A computational procedure for incomplete gamma functions, Rend. Sem. Mat. Univ. Politec. Torino 37 (1979), no. 1, 1–9 (Italian). MR 547763
  • W. Gautschi, "Algorithm 542—Incomplete gamma function," ACM Trans. Math. Software, v. 5, 1979, pp. 482-489.
  • Peter Henrici, Applied and computational complex analysis. Vol. 2, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977. Special functions—integral transforms—asymptotics—continued fractions. MR 0453984
  • E. P. Merkes, On truncation errors for continued fraction computations, SIAM J. Numer. Anal. 3 (1966), 486–496. MR 202283, DOI 10.1137/0703042
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Additional Information
  • © Copyright 1983 American Mathematical Society
  • Journal: Math. Comp. 40 (1983), 337-342
  • MSC: Primary 40A15; Secondary 33A15
  • DOI:
  • MathSciNet review: 679450