In mathematics, trigonometric integrals are a family of nonelementary 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. [1]
Various expansions can be used for evaluation of trigonometric integrals, depending on the range of the argument.
These series are asymptotic and divergent, although can be used for estimates and even precise evaluation at ℜ(x) ≫ 1.
These series are convergent at any complex x, although for |x| ≫ 1, the series will converge slowly initially, requiring many terms for high precision.
From the Maclaurin series expansion of sine:
The function
is called the exponential integral. It is closely related to Si and Ci,
As each respective function is analytic except for the cut at negative values of the argument, the area of validity of the relation should be extended to (Outside this range, additional terms which are integer factors of π appear in the expression.)
Cases of imaginary argument of the generalized integro-exponential function are
which is the real part of
Similarly
Padé approximants of the convergent Taylor series provide an efficient way to evaluate the functions for small arguments. The following formulae, given by Rowe et al. (2015), [2] are accurate to better than 10−16 for 0 ≤ x ≤ 4,
The integrals may be evaluated indirectly via auxiliary functions and , which are defined by
or equivalently | ||
For the Padé rational functions given below approximate and with error less than 10−16: [2]
In mathematics, the logarithmic integral function or integral logarithm li(x) is a special function. It is relevant in problems of physics and has number theoretic significance. In particular, according to the prime number theorem, it is a very good approximation to the prime-counting function, which is defined as the number of prime numbers less than or equal to a given value .
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the unit hyperbola. Also, similarly to how the derivatives of sin(t) and cos(t) are cos(t) and –sin(t) respectively, the derivatives of sinh(t) and cosh(t) are cosh(t) and +sinh(t) respectively.
In complex analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as well. It generalizes the Cauchy integral theorem and Cauchy's integral formula. The residue theorem should not be confused with special cases of the generalized Stokes' theorem; however, the latter can be used as an ingredient of its proof.
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. It is frequently used to transform the antiderivative of a product of functions into an antiderivative for which a solution can be more easily found. The rule can be thought of as an integral version of the product rule of differentiation; it is indeed derived using the product rule.
Integration is the basic operation in integral calculus. While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful. This page lists some of the most common antiderivatives.
The Fresnel integralsS(x) and C(x) are two transcendental functions named after Augustin-Jean Fresnel that are used in optics and are closely related to the error function (erf). They arise in the description of near-field Fresnel diffraction phenomena and are defined through the following integral representations:
In the physical sciences, the Airy function (or Airy function of the first kind) Ai(x) is a special function named after the British astronomer George Biddell Airy (1801–1892). The function Ai(x) and the related function Bi(x), are linearly independent solutions to the differential equation
In mathematics, the Clausen function, introduced by Thomas Clausen, is a transcendental, special function of a single variable. It can variously be expressed in the form of a definite integral, a trigonometric series, and various other forms. It is intimately connected with the polylogarithm, inverse tangent integral, polygamma function, Riemann zeta function, Dirichlet eta function, and Dirichlet beta function.
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:
In mathematics, physics and engineering, the sinc function, denoted by sinc(x), has two forms, normalized and unnormalized.
In mathematics, the exponential integral Ei is a special function on the complex plane.
In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.
In mathematics, there are several integrals known as the Dirichlet integral, after the German mathematician Peter Gustav Lejeune Dirichlet, one of which is the improper integral of the sinc function over the positive real line:
In mathematics, the Stieltjes constants are the numbers that occur in the Laurent series expansion of the Riemann zeta function:
In mathematics, the reciprocal gamma function is the function
Carl Johan Malmsten was a Swedish mathematician and politician. He is notable for early research into the theory of functions of a complex variable, for the evaluation of several important logarithmic integrals and series, for his studies in the theory of Zeta-function related series and integrals, as well as for helping Mittag-Leffler start the journal Acta Mathematica. Malmsten became Docent in 1840, and then, Professor of mathematics at the Uppsala University in 1842. He was elected a member of the Royal Swedish Academy of Sciences in 1844. He was also a minister without portfolio in 1859–1866 and Governor of Skaraborg County in 1866–1879.
In mathematics, a Borwein integral is an integral whose unusual properties were first presented by mathematicians David Borwein and Jonathan Borwein in 2001. Borwein integrals involve products of , where the sinc function is given by for not equal to 0, and .
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