Pi: A Source Book pp 448-455 | Cite as

# Some New Algorithms for High-Precision Computation of Euler’s Constant

Chapter

## Abstract

We describe several new algorithms for the high-precision computation of Euler’s constant *γ* = 0.577.... Using one of the algorithms, which is based on an identity involving Bessel functions, *γ* has been computed to 30,100 decimal places. By computing their regular continued fractions we show that, if *γ* or exp(*γ*) is of the form *P/Q* for integers *P* and *Q*, then |*Q*| *>*10^{15000}

## Keywords

Asymptotic Expansion Continue Fraction Partial Quotient Precision Number Involve Bessel Function
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.M. Abramowitz & I. A. Stegun,
*handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables*, National Bureau of Standards, Washington, D. C., 1964.Google Scholar - 2.M. Beeler, R. W. Gosielt & R. Sciiroeilkl, “Ilakmem,” Memo No. 239, Artificial Intelligence Lab., Cambridge, Mass., 1972, pp. 70–71.Google Scholar
- 3.W. A. Beyer & M. S. Waterman, “Error analysis of a computation of Euler’s constant,”
*Math. Comp*.,*v*. 28, 1974, pp. 599–604. MR**49**#6555.Google Scholar - W. A. Beyer & M. S. Waterman, “Decimals and partial quotients of Euler’s constant and In 2,” UMT
**19***Math. Comp*.,*v*. 28, 1974, p. 667. Errata:*Math. Comp*.,MTE**549**, v. 32, 1978, pp. 317–318.Google Scholar - 5.R. P. Brent, “The complexity of multiple-precision arithmetic,”
*Complexity of Computational Problem Solving*(R. S. Anderssen and R. P. Brent, Eds.), Univ. of Queensland Press, Brisbane, 1976, pp. 126–165.Google Scholar - 6.R. P. Brent, “Multiple-precision zero-finding methods and the complexity of elementary function evaluation,”
*Analytic Computational Complexity*(J. F. Traub, Ed.), Academic Press, New York, 1976, pp. 151–176. MR**52**#15938,**54**#11843.Google Scholar - 7.R. P. Brent, “Computation of the regular continued fraction for Euler’s constant,”Math. Comp., v. 31, 1977, pp. 771–777. MR
**55**#9490.Google Scholar - 8.R. P. Brent, “?’ and exp(y) to 20700D and their regular continued fractions to 20000 partial quotients,” UMT 1,
*Math. Co*,*np*.,*v*. 32, 1978, p. 311.Google Scholar - 9.R. l. Brent, “A Fortran multiple-precision arithmetic package,” ACM Trans. Math. Software, v. 4, 1978,
_{pp.}57–70.Google Scholar - 10.R. P. Brent, “Euler’s constant and its exponential to 30,100 decimals,” Computing Research Group, Australian National University, Sept. 1978. Submitted to
*Math. Comp*. UMT file.Google Scholar - 11.R. P. Brent & E. M. Mcmillan, “The first 29,000 partial quotients in the regular continued fractions for Euler’s constant and its exponential,” submitted
*to Math. Comp. UMT file*.Google Scholar - 12.J. W. L. Glaisher, “History of Euler’s constant,”
*Messenger of Math*.,*v*. 1, 1872, pp. 25–30.Google Scholar - 13.A. YA. Khintci-Line (A. JA. Hingin),
*Continued Fractions*,3rd ed., (English transl. by P. Wynn), Noordhoff, Groningen, 1963. MR 28 #5038.Google Scholar - 14.G. F. B. Riemann, “Zur Theorie der Nobili’schen Farbenringe,”
*Poggendorff’s Annalen der Physik und Chemie*, Bd. 95, 1855, pp. 130–139. (Reprinted in*Bernhard Riemann’s Gesammelte Mathematische Werke und Wissenschaftlicher Nachlass*Teubner, Leipzig, 1876, pp. 54–61.)Google Scholar - 15.A. Schonhage & V. Strassen, “Schnelle Multiplikation grosser Zahlen,”
*Computing*,*v*.*7*, 1971, pp. 281–292.MathSciNetCrossRefGoogle Scholar - 16.D. W. Sweeney, “On the computation of Euler’s constant,”
*Math. Comp*.,*v*. 17, 1963, pp. 170–178. MR**28**#3522.Google Scholar - 17.A. Van Der Poorten, “A proof that Euler missed—Apéry’s proof of the irrationality of j’(3),”
*Mathematical Intelligence*,*v*. 1, 1979, pp. 196–203.Google Scholar - 18.
- 19.G. N. Watson,
*A Treatise on the Theory of Bessel Functions*, 2nd ed., Cambridge Univ. Press, London, 1944.MATHGoogle Scholar

## Copyright information

© Springer Science+Business Media New York 2000