Trigonometric 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.

Sine integral

The different sine integral definitions are

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 = ∞. Their difference is given by the Dirichlet integral,

Note that sin x / x is the sinc function, and also the zeroth spherical Bessel function.

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.

Related is the Gibbs phenomenon: if the sine integral is considered as the convolution of the sinc function with the heaviside step function, this corresponds to truncating the Fourier series, which is the cause of the Gibbs phenomenon.