Definition:Du Bois-Reymond Constants

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Definition

The du Bois-Reymond constants are the constants $C_n$ where:

$C_n = \displaystyle \int_0^\infty \size {\map {\dfrac \d {\d t} } {\dfrac {\sin t} t}^n} \rd t - 1$


Examples

First du Bois-Reymond Constant

The first du Bois-Reymond constant $C_1$ does not exist.

This is because:

$\displaystyle \int_0^\infty \size {\map {\dfrac \d {\d t} } {\dfrac {\sin t} t}^n} \rd t - 1$

does not converge.


Second du Bois-Reymond Constant

The second du Bois-Reymond constant $C_2$ evaluates as:

\(\displaystyle C_2\) \(=\) \(\displaystyle \dfrac {e^2 - 7} 2\)
\(\displaystyle \) \(\approx\) \(\displaystyle 0 \cdotp 19452 \, 80494 \, 6532 \ldots\)


Source of Name

This entry was named for Paul David Gustav du Bois-Reymond‎.


Sources