Sine Integral
The different sine integral definitions are:
is the primitive of which is zero for ; is the primitive of which is zero for .
Note that is the sinc function and also the zeroth spherical Bessel function.
When, this is known as the Dirichlet integral.
In signal processing, the oscillations of the Sine integral cause overshoot and ringing artifacts when using the sinc filter, and frequency domain ringing if using a truncated sinc filter as a low-pass filter.
The Gibbs phenomenon is a related phenomenon: thinking of sinc as a low-pass filter, it corresponds to truncating the Fourier series, which causes the Gibbs phenomenon.
Read more about this topic: Trigonometric Integral
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