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# du Bois-Reymond Constants

The constants defined by

 (1)

These constants can also be written as the sums

 (2)

and

 (3)

(E. Weisstein, Feb. 3, 2015), where is the th positive root of

 (4)

and is the sinc function.

diverges, with the first few subsequent constant numerically given by

 (5) (6) (7)

Rather surprisingly, the even-ordered du Bois Reymond constants (and, in particular, ; Le Lionnais 1983) can be computed analytically as polynomials in ,

 (8) (9) (10)

(OEIS A085466 and A085467) as found by Watson (1933). For positive integer , these have the explicit formula

 (11)

where denotes a complex residue and is a Kronecker delta (V. Adamchik).

Series, Tanc Function

## References

Finch, S. R. "Du Bois Reymond's Constants." §3.12 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 237-240, 2003.Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 23, 1983.Sloane, N. J. A. Sequences A085466 and A085467 in "The On-Line Encyclopedia of Integer Sequences."Watson, G. N. "Du Bois Reymond's Constants." Quart. J. ath. 4, 140-146, 1933.Young, R. M. "A Rayleigh Popular Problem." Amer. Math. Monthly 93, 660-664, 1986.

## Cite this as:

Weisstein, Eric W. "du Bois-Reymond Constants." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/duBois-ReymondConstants.html