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
Famous quotes containing the words sine and/or integral:
“Faultless honesty is a sine qua non of business life. Not alone the honesty according to the moral code and the Bible. When I speak of honesty I refer to the small, hidden, evasive meannesses of our natures. I speak of the honesty of ourselves to ourselves.”
—Alice Foote MacDougall (1867–1945)
“... no one who has not been an integral part of a slaveholding community, can have any idea of its abominations.... even were slavery no curse to its victims, the exercise of arbitrary power works such fearful ruin upon the hearts of slaveholders, that I should feel impelled to labor and pray for its overthrow with my last energies and latest breath.”
—Angelina Grimké (1805–1879)