Sine integral

In mathematics, the trigonometric integrals are a family of integrals involving trigonometric functions. A number of the basic trigonometric integrals are discussed at the list of integrals of trigonometric functions.

The different sine integral definitions are

Note that the integrand $sin x / x$ is the $sinc$ function, and also the zeroth spherical Bessel function. Since $sinc$ is an even entire function (holomorphic over the entire complex plane), $Si$ is entire, odd, and the integral in its definition can be taken along any path connecting the endpoints.

By definition, $Si(x)$ is the antiderivative of $sin x / x$ which is zero for $x = 0$ , and $si(x)$ is the antiderivative of $sin x / x$ which is zero for $x = \infty$ . Their difference is given by 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.

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