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Wilbraham-Gibbs constant
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Numbers
$G'$:
1.851937051982466170361053370157991363345809728981154909804783781876981890166348358532710336502954757
$G$:
1.178979744472167270232028845824909741463897420964366146834503705768303703705043859077668347949410421
$G-1$:
0.178979744472167270232028845824909741463897420964366146834503705768303703705043859077668347949410421
$(G-1)/2$:
0.089489872236083635116014422912454870731948710482183073417251852884151851852521929538834173974705210
Definition
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).
Links
Data properties
Entries are of type: real number