# Definition:Du Bois-Reymond Constants

## 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.