Trigonometry/The Gibbs Overshoot

On the previous page, it can be seen that the approximations to the square wave go slightly above the wave, and then come down again. As more terms are added, this overshoot gets slightly greater but persists for a smaller range of values of x before dropping back to the right level. In the limit, as the number of terms tends to infinity, the maximum value reaches about 9% more than the level of the square wave. The precise value is

${\frac {2}{\pi }}\int \limits _{0}^{\pi }{\frac {\sin(t)}{t}}\,dt\approx 1.089490$

This overshoot is known as the Gibbs effect, Gibbs phenomenon or Gibbs overshoot, after the mathematical physicist Josiah Gibbs (1839-1903), who explained the phenomenon in 1899.