Trigonometric integral

In mathematics, the trigonometric integrals are a family of integrals involving 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 whose value is zero at zero of sin x / x for x = 0, and si(x) is the antiderivative whose value at x = ∞ is zero of sin x / x. 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. 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. The different cosine integral definitions are where γ is the Euler–Mascheroni constant. Some texts use ci instead of Ci.