info prev up next book cdrom email home

Du Bois Raymond Constants

\begin{figure}\begin{center}\BoxedEPSF{duBoisRaymondConstants.epsf}\end{center}\end{figure}

The constants $C_n$ defined by

\begin{displaymath} C_n\equiv \int_0^\infty \left\vert{{d\over dt}\left({\sin t\over t}\right)^n}\right\vert\,dt-1 \end{displaymath}

which are difficult to compute numerically. The first few are
$\displaystyle C_1$ $\textstyle \approx$ $\displaystyle 455$  
$\displaystyle C_2$ $\textstyle \approx$ $\displaystyle 0.1945$  
$\displaystyle C_3$ $\textstyle \approx$ $\displaystyle 0.028254$  
$\displaystyle C_4$ $\textstyle \approx$ $\displaystyle 0.00524054.$  

Rather surprisingly, the second Du Bois Raymond constant is given analytically by

\begin{displaymath} C_2={\textstyle{1\over 2}}(e^2-7)=0.1945280494\ldots \end{displaymath}

(Le Lionnais 1983).


References

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 23, 1983.

Plouffe, S. ``Dubois-Raymond 2nd Constant.'' http://www.lacim.uqam.ca/piDATA/dubois.txt.



© 1996-9 Eric W. Weisstein
1999-05-24