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du Bois-Reymond Constants

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👁 duBoisReymondConstants

The constants 👁 C_n
defined by

👁 C_n=[int_0^infty|d/(dt)((sint)/t)^n|dt]-1.

(1)

These constants can also be written as the sums

👁 C_n=2sum_(k=1)^infty(1+x_k^2)^(-n/2),

(2)

and

👁 C_n=2sum_(k=1)^infty|sinc(x_k)|^n

(3)

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

👁 t=tant

(4)

and 👁 sinc(x)
is the sinc function.

👁 C_1
diverges, with the first few subsequent constant numerically given by

👁 C_2	👁 approx	👁 0.1945280494	(5)
👁 C_3	👁 approx	👁 0.02825176416	(6)
👁 C_4	👁 approx	👁 0.005240704678.	(7)

Rather surprisingly, the even-ordered du Bois Reymond constants (and, in particular, 👁 C_2
; Le Lionnais 1983) can be computed analytically as polynomials in 👁 e^2
,

👁 C_2	👁 =	👁 1/2(e^2-7)	(8)
👁 C_4	👁 =	👁 1/8(e^4-4e^2-25)	(9)
👁 C_6	👁 =	👁 1/(32)(e^6-6e^4+3e^2-98)	(10)

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

👁 C_(2n)=-(3+delta_(1n))-2Res_(x=i)[(x^2)/((1+x^2)^n(tanx-x))],

(11)

where 👁 Res
denotes a complex residue and 👁 delta_(ij)
is a Kronecker delta (V. Adamchik).

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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.

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du Bois-Reymond Constants

Cite this as:

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

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