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Du Bois Raymond Constants


The constants $C_n$ defined by

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

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

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

(Le Lionnais 1983).


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

Plouffe, S. ``Dubois-Raymond 2nd Constant.''

© 1996-9 Eric W. Weisstein