The Wilbraham-Gibbs constant $G'$ is defined as $G' = \text{Si}(\pi) = \int_0^\pi \frac{\sin t}{t} dt$. The Gibbs constant $G$ is defined as $G'/(\pi/2)$.
Comments
(1)
If $f$ is a periodic piecewise continuously differentiable function with a jump of a certain size at a discontinuity of $f$, then for large $n$, the $n$'th partial Fourier expansion of $f$ will overshoot this jump by a factor of approximately $(G-1)/2$ at each side, and by a factor of $(G-1)$ in total. This is known as the Gibbs phenomenon [6].
References
[1]
Finch, S. R., "Gibbs-Wilbraham Constant." §4.1 in "Mathematical Constants". Cambridge University Press, pp. 248-250, (2003).
[2]
Wilbraham, Henry, "On a certain periodic function", The Cambridge and Dublin Mathematical Journal, 3: 198–201, (1848).
How they were obtained: Computed in interval or ball arithmetic throughout, so the digits written follow from the width of the result rather than from a guard chosen in advance.
