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Concept
Trigonometric integral
Summary
In mathematics, trigonometric integrals are a family of integrals involving trigonometric functions.
The different sine integral definitions are
Note that the integrand 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 whose value is zero at x = 0, and si(x) is the antiderivative whose value is zero at 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 ≈ 0.57721566 ... is the Euler–Mascheroni constant. Some texts use ci instead of Ci.
Ci(x) is the antiderivative of cos x / x (which vanishes as ). The two definitions are related by
Cin is an even, entire function. For that reason, some texts treat Cin as the primary function, and derive Ci in terms of Cin.
The hyperbolic sine integral is defined as
It is related to the ordinary sine integral by
The hyperbolic cosine integral is
where is the Euler–Mascheroni constant.
It has the series expansion
Trigonometric integrals can be understood in terms of the so-called "auxiliary functions"
Using these functions, the trigonometric integrals may be re-expressed as
(cf. Abramowitz & Stegun, p. 232)
The spiral formed by parametric plot of si , ci is known as Nielsen's spiral.
The spiral is closely related to the Fresnel integrals and the Euler spiral. Nielsen's spiral has applications in vision processing, road and track construction and other areas.
Official source
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