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.
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Ce cours pose les bases d'un concept essentiel en ingénierie : la notion de système. Plus spécifiquement, le cours présente la théorie des systèmes linéaires invariants dans le temps (SLIT), qui sont
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Adaptive signal processing, A/D and D/A. This module provides the basic
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In mathematics, the Gibbs phenomenon is the oscillatory behavior of the Fourier series of a piecewise continuously differentiable periodic function around a jump discontinuity. The th partial Fourier series of the function (formed by summing the lowest constituent sinusoids of the Fourier series of the function) produces large peaks around the jump which overshoot and undershoot the function values.
In mathematics, physics and engineering, the sinc function, denoted by sinc(x), has two forms, normalized and unnormalized. In mathematics, the historical unnormalized sinc function is defined for x ≠ 0 by Alternatively, the unnormalized sinc function is often called the sampling function, indicated as Sa(x). In digital signal processing and information theory, the normalized sinc function is commonly defined for x ≠ 0 by In either case, the value at x = 0 is defined to be the limiting value for all real a ≠ 0 (the limit can be proven using the squeeze theorem).
In signal processing, particularly , ringing artifacts are artifacts that appear as spurious signals near sharp transitions in a signal. Visually, they appear as bands or "ghosts" near edges; audibly, they appear as "echos" near transients, particularly sounds from percussion instruments; most noticeable are the pre-echos. The term "ringing" is because the output signal oscillates at a fading rate around a sharp transition in the input, similar to a bell after being struck.
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