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On Euler’s attempt to compute logarithms by interpolation: a commentary to his letter of February 16, 1734 to Daniel Bernoulli. (English) Zbl 1146.01004

A letter from Euler to D. Bernoulli described a function \(\sum_{n=0}^\infty c_n\prod_{k=0}^n(x-10^k)\), with the coefficients chosen to make the function agree with the (base-10) logarithm whenever \(x\) is a nonnegative power of ten. Euler had hoped thereby to represent the logarithm, but he noted that evaluating the series at \(x=9\) yields the wrong answer. The present paper analyzes Euler’s function and its generalization to bases other than 10 as a \(q\)-analog of the logarithm. The author then shows how these other-based functions may be used to calculate base-10 logs. Numerical issues such as roundoff and cancellation errors and rapidity of convergence are also considered.

MSC:

01A50 History of mathematics in the 18th century
65-03 History of numerical analysis

Biographic References:

Euler, Leonhard

Software:

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Full Text: DOI

References:

[1] Andrews, G. E.; Askey, R.; Roy, R., Special functions, Encyclopedia of Mathematics and its Applications, vol. 71 (1999), Cambridge University Press: Cambridge University Press Cambridge
[2] G. Eneström, Bib. Math. 7(3) 1906-1907, 134-137.; G. Eneström, Bib. Math. 7(3) 1906-1907, 134-137.
[3] L. Euler, Consideratio quarumdam serierum quae singularibus proprietatibus sunt praeditae, Novi Comment. Acad. Sci. Petropolitanae 3 (1750/51) 1753, 10-12, 86-108. (Also in Leonhardi Euleri Opera Omnia, Ser. I, vol. 14, pp. 516-541, B.G. Teubner, Leipzig and Berlin, 1925. An English translation of this memoir can be downloaded from the E190 page of the Euler Archive at http://www.math.dartmouth.edu/\( \sim;\) euler.); L. Euler, Consideratio quarumdam serierum quae singularibus proprietatibus sunt praeditae, Novi Comment. Acad. Sci. Petropolitanae 3 (1750/51) 1753, 10-12, 86-108. (Also in Leonhardi Euleri Opera Omnia, Ser. I, vol. 14, pp. 516-541, B.G. Teubner, Leipzig and Berlin, 1925. An English translation of this memoir can be downloaded from the E190 page of the Euler Archive at http://www.math.dartmouth.edu/\( \sim;\) euler.)
[4] Gautschi, W., Numerical Analysis: An Introduction (1997), Birkhäuser: Birkhäuser Boston, MA · Zbl 0877.65001
[5] E. Koelink, W. Van Assche, Leonhard Euler and a \(q\)-analogue of the logarithm, in preparation.; E. Koelink, W. Van Assche, Leonhard Euler and a \(q\)-analogue of the logarithm, in preparation. · Zbl 1188.33006
[6] Muller, J. M., Elementary Functions: Algorithms and Implementation (2006), Birkhäuser: Birkhäuser Boston, MA · Zbl 1089.65016
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