Trigonometric integral
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In mathematics, trigonometric integrals are a family of nonelementary integrals involving trigonometric functions.
Sine integral
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The different sine integral definitions are <math display="block">\operatorname{Si}(x) = \int_0^x\frac{\sin t}{t}\,dt</math> <math display="block">\operatorname{si}(x) = -\int_x^\infty\frac{\sin t}{t}\,dt~.</math>
Note that the integrand <math>\frac{\sin(t)}{t}</math> is the sinc function, and also the zeroth spherical Bessel function. Since Template:Math is an even entire function (holomorphic over the entire complex plane), Template:Math is entire, odd, and the integral in its definition can be taken along any path connecting the endpoints.
By definition, Template:Math is the antiderivative of Template:Math whose value is zero at Template:Math, and Template:Math is the antiderivative whose value is zero at Template:Math. Their difference is given by the Dirichlet integral, <math display="block">\operatorname{Si}(x) - \operatorname{si}(x) = \int_0^\infty\frac{\sin t}{t}\,dt = \frac{\pi}{2} \quad \text{ or } \quad \operatorname{Si}(x) = \frac{\pi}{2} + \operatorname{si}(x) ~.</math>
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.
Cosine integral
The different cosine integral definitions are <math display="block">\operatorname{Cin}(x) ~\equiv~ \int_0^x \frac{\ 1 - \cos t\ }{ t }\ \operatorname{d} t ~.</math>
Template:Math is an even, entire function. For that reason, some texts define Template:Math as the primary function, and derive Template:Math in terms of Template:Math
<math display="block">\operatorname{Ci}(x) ~~\equiv~ -\int_x^\infty \frac{\ \cos t\ }{ t }\ \operatorname{d} t ~</math> <math display="block">~~ \qquad ~=~~ \gamma ~+~ \ln x ~-~ \int_0^x \frac{\ 1 - \cos t\ }{ t }\ \operatorname{d} t ~</math>
<math display="block">~~ \qquad ~=~~ \gamma ~+~ \ln x ~-~ \operatorname{Cin} x ~</math> for <math>~\Bigl|\ \operatorname{Arg}(x)\ \Bigr| < \pi\ ,</math> where Template:Math is the Euler–Mascheroni constant. Some texts use Template:Math instead of Template:Math. The restriction on Template:Math is to avoid a discontinuity (shown as the orange vs blue area on the left half of the plot above) that arises because of a branch cut in the standard logarithm function (Template:Math).
Template:Math is the antiderivative of Template:Math (which vanishes as <math>\ x \to \infty\ </math>). The two definitions are related by <math display="block">\operatorname{Ci}(x) = \gamma + \ln x - \operatorname{Cin}(x) ~.</math>
Hyperbolic sine integral
The hyperbolic sine integral is defined as <math display="block">\operatorname{Shi}(x) =\int_0^x \frac {\sinh (t)}{t}\,dt.</math>
It is related to the ordinary sine integral by <math display="block">\operatorname{Si}(ix) = i\operatorname{Shi}(x).</math>
Hyperbolic cosine integral
The hyperbolic cosine integral is
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<math display="block">\operatorname{Chi}(x) = \gamma+\ln x + \int_0^x\frac{\cosh t-1}{t}\,dt \qquad ~ \text{ for } ~ \left| \operatorname{Arg}(x) \right| < \pi~,</math> where <math>\gamma</math> is the Euler–Mascheroni constant.
It has the series expansion <math display="block">\operatorname{Chi}(x) = \gamma + \ln(x) + \frac {x^2}{4} + \frac {x^4}{96} + \frac {x^6}{4320} + \frac {x^8}{322560} + \frac{x^{10}}{36288000} + O(x^{12}).</math>
Auxiliary functions
Trigonometric integrals can be understood in terms of the so-called "auxiliary functions" <math display="block"> \begin{array}{rcl} f(x) &\equiv& \int_0^\infty \frac{\sin(t)}{t+x} \,dt &=& \int_0^\infty \frac{e^{-x t}}{t^2 + 1} \,dt
&=& \operatorname{Ci}(x) \sin(x) + \left[\frac{\pi}{2} - \operatorname{Si}(x) \right] \cos(x)~, \\
g(x) &\equiv& \int_0^\infty \frac{\cos(t)}{t+x} \,dt &=& \int_0^\infty \frac{t e^{-x t}}{t^2 + 1} \,dt
&=& -\operatorname{Ci}(x) \cos(x) + \left[\frac{\pi}{2} - \operatorname{Si}(x) \right] \sin(x)~.
\end{array} </math> Using these functions, the trigonometric integrals may be re-expressed as (cf. Abramowitz & Stegun, p. 232) <math display="block">\begin{array}{rcl} \frac{\pi}{2} - \operatorname{Si}(x) = -\operatorname{si}(x) &=& f(x) \cos(x) + g(x) \sin(x)~, \qquad \text{ and } \\ \operatorname{Ci}(x) &=& f(x) \sin(x) - g(x) \cos(x)~. \\ \end{array}</math>
Nielsen's spiral
The spiral formed by parametric plot of Template:Math is known as Nielsen's spiral. <math display="block">x(t) = a \times \operatorname{ci}(t)</math> <math display="block">y(t) = a \times \operatorname{si}(t)</math>
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.<ref>Template:Cite book</ref>
Expansion
Various expansions can be used for evaluation of trigonometric integrals, depending on the range of the argument.
Asymptotic series (for large argument)
<math display="block">\operatorname{Si}(x) \sim \frac{\pi}{2}
- \frac{\cos x}{x}\left(1-\frac{2!}{x^2}+\frac{4!}{x^4}-\frac{6!}{x^6}\cdots\right)
- \frac{\sin x}{x}\left(\frac{1}{x}-\frac{3!}{x^3}+\frac{5!}{x^5}-\frac{7!}{x^7}\cdots\right)</math>
<math display="block">\operatorname{Ci}(x) \sim \frac{\sin x}{x}\left(1-\frac{2!}{x^2}+\frac{4!}{x^4}-\frac{6!}{x^6}\cdots\right)
- \frac{\cos x}{x}\left(\frac{1}{x}-\frac{3!}{x^{3}}+\frac{5!}{x^5}-\frac{7!}{x^7}\cdots\right) ~.</math>
These series are asymptotic and divergent, although can be used for estimates and even precise evaluation at Template:Math.
Convergent series
<math display="block">\operatorname{Si}(x)= \sum_{n=0}^\infty \frac{(-1)^{n}x^{2n+1}}{(2n+1)(2n+1)!}=x-\frac{x^3}{3!\cdot3}+\frac{x^5}{5!\cdot5}-\frac{x^7}{7! \cdot7}\pm\cdots</math> <math display="block">\operatorname{Ci}(x)= \gamma+\ln x+\sum_{n=1}^{\infty}\frac{(-1)^{n}x^{2n}}{2n(2n)!}=\gamma+\ln x-\frac{x^2}{2!\cdot2} + \frac{x^4}{4! \cdot4}\mp\cdots</math>
These series are convergent at any complex Template:Mvar, although for Template:Math, the series will converge slowly initially, requiring many terms for high precision.
Derivation of series expansion
From the Maclaurin series expansion of sine: <math display="block">\sin\,x = x - \frac{x^3}{3!}+\frac{x^5}{5!}- \frac{x^7}{7!}+\frac{x^9}{9!}-\frac{x^{11}}{11!} + \cdots</math> <math display="block">\frac{\sin\,x}{x} = 1 - \frac{x^2}{3!}+\frac{x^4}{5!}- \frac{x^6}{7!}+\frac{x^8}{9!}-\frac{x^{10}}{11!}+\cdots</math> <math display="block">\therefore\int \frac{\sin\,x}{x}dx = x - \frac{x^3}{3!\cdot3}+\frac{x^5}{5!\cdot5}- \frac{x^7}{7!\cdot7}+\frac{x^9}{9!\cdot9}-\frac{x^{11}}{11!\cdot11}+\cdots </math>
Relation with the exponential integral of imaginary argument
The function <math display="block">\operatorname{E}_1(z) = \int_1^\infty \frac{\exp(-zt)}{t}\,dt \qquad~\text{ for }~ \Re(z) \ge 0 </math> is called the exponential integral. It is closely related to Template:Math and Template:Math, <math display="block"> \operatorname{E}_1(i x) = i\left(-\frac{\pi}{2} + \operatorname{Si}(x)\right)-\operatorname{Ci}(x) = i \operatorname{si}(x) - \operatorname{Ci}(x) \qquad ~\text{ for }~ x > 0 ~. </math>
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 Template:Math appear in the expression.)
Cases of imaginary argument of the generalized integro-exponential function are <math display="block"> \int_1^\infty \cos(ax)\frac{\ln x}{x} \, dx = -\frac{\pi^2}{24}+\gamma\left(\frac{\gamma}{2}+\ln a\right)+\frac{\ln^2a}{2} +\sum_{n\ge 1} \frac{(-a^2)^n}{(2n)!(2n)^2} ~, </math> which is the real part of <math display="block"> \int_1^\infty e^{iax}\frac{\ln x}{x}\,dx = -\frac{\pi^2}{24} + \gamma\left(\frac{\gamma}{2}+\ln a\right)+\frac{\ln^2 a}{2} -\frac{\pi}{2}i\left(\gamma+\ln a\right) + \sum_{n\ge 1}\frac{(ia)^n}{n!n^2} ~. </math>
Similarly <math display="block"> \int_1^\infty e^{iax}\frac{\ln x}{x^2}\,dx =
1 + ia\left[ -\frac{\pi^2}{24} + \gamma \left( \frac{\gamma}{2} + \ln a - 1 \right) + \frac{\ln^2 a}{2} - \ln a + 1 \right]
+ \frac{\pi a}{2} \Bigl( \gamma+\ln a - 1 \Bigr)
+ \sum_{n\ge 1}\frac{(ia)^{n+1}}{(n+1)!n^2}~.
</math>
Efficient evaluation
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),<ref name=RoweEtAl2015>Template:Cite journal</ref> are accurate to better than Template:Math for Template:Math, <math display="block">\begin{array}{rcl} \operatorname{Si}(x) &\approx & x \cdot \left( \frac{ \begin{array}{l} 1 -4.54393409816329991\cdot 10^{-2} \cdot x^2 + 1.15457225751016682\cdot 10^{-3} \cdot x^4 - 1.41018536821330254\cdot 10^{-5} \cdot x^6 \\ ~~~ + 9.43280809438713025 \cdot 10^{-8} \cdot x^8 - 3.53201978997168357 \cdot 10^{-10} \cdot x^{10} + 7.08240282274875911 \cdot 10^{-13} \cdot x^{12} \\ ~~~ - 6.05338212010422477 \cdot 10^{-16} \cdot x^{14} \end{array} } { \begin{array}{l} 1 + 1.01162145739225565 \cdot 10^{-2} \cdot x^2 + 4.99175116169755106 \cdot 10^{-5} \cdot x^4 + 1.55654986308745614 \cdot 10^{-7} \cdot x^6 \\ ~~~ + 3.28067571055789734 \cdot 10^{-10} \cdot x^8 + 4.5049097575386581 \cdot 10^{-13} \cdot x^{10} + 3.21107051193712168 \cdot 10^{-16} \cdot x^{12} \end{array} } \right)\\ &~&\\ \operatorname{Ci}(x) &\approx & \gamma + \ln(x) +\\ && x^2 \cdot \left( \frac{ \begin{array}{l} -0.25 + 7.51851524438898291 \cdot 10^{-3} \cdot x^2 - 1.27528342240267686 \cdot 10^{-4} \cdot x^4 + 1.05297363846239184 \cdot 10^{-6} \cdot x^6 \\ ~~~ -4.68889508144848019 \cdot 10^{-9} \cdot x^8 + 1.06480802891189243 \cdot 10^{-11} \cdot x^{10} - 9.93728488857585407 \cdot 10^{-15} \cdot x^{12} \\ \end{array} } { \begin{array}{l} 1 + 1.1592605689110735 \cdot 10^{-2} \cdot x^2 + 6.72126800814254432 \cdot 10^{-5} \cdot x^4 + 2.55533277086129636 \cdot 10^{-7} \cdot x^6 \\ ~~~ + 6.97071295760958946 \cdot 10^{-10} \cdot x^8 + 1.38536352772778619 \cdot 10^{-12} \cdot x^{10} + 1.89106054713059759 \cdot 10^{-15} \cdot x^{12} \\ ~~~ + 1.39759616731376855 \cdot 10^{-18} \cdot x^{14} \\ \end{array} } \right) \end{array}</math>
The integrals may be evaluated indirectly via auxiliary functions <math>f(x)</math> and <math>g(x)</math>, which are defined by
| <math display="block">\operatorname{Si}(x)=\frac{\pi}{2}-f(x)\cos(x)-g(x)\sin(x)</math> | <math display="block">\operatorname{Ci}(x)=f(x)\sin(x)-g(x)\cos(x) </math> | |
| or equivalently | ||
| <math display="block">f(x) \equiv \left[\frac{\pi}{2} - \operatorname{Si}(x)\right] \cos(x) + \operatorname{Ci}(x) \sin(x)</math> | <math display="block">g(x) \equiv \left[\frac{\pi}{2} - \operatorname{Si}(x)\right] \sin(x) - \operatorname{Ci}(x) \cos(x)</math> | |
For <math>x \ge 4</math> the Padé rational functions given below approximate <math>f(x)</math> and <math>g(x)</math> with error less than 10−16:<ref name=RoweEtAl2015/>
<math display="block">\begin{array}{rcl} f(x) &\approx & \dfrac{1}{x} \cdot \left(\frac{ \begin{array}{l} 1 + 7.44437068161936700618 \cdot 10^2 \cdot x^{-2} + 1.96396372895146869801 \cdot 10^5 \cdot x^{-4} + 2.37750310125431834034 \cdot 10^7 \cdot x^{-6} \\ ~~~ + 1.43073403821274636888 \cdot 10^9 \cdot x^{-8} + 4.33736238870432522765 \cdot 10^{10} \cdot x^{-10} + 6.40533830574022022911 \cdot 10^{11} \cdot x^{-12} \\ ~~~ + 4.20968180571076940208 \cdot 10^{12} \cdot x^{-14} + 1.00795182980368574617 \cdot 10^{13} \cdot x^{-16} + 4.94816688199951963482 \cdot 10^{12} \cdot x^{-18} \\ ~~~ - 4.94701168645415959931 \cdot 10^{11} \cdot x^{-20} \end{array} }{ \begin{array}{l} 1 + 7.46437068161927678031 \cdot 10^2 \cdot x^{-2} + 1.97865247031583951450 \cdot 10^5 \cdot x^{-4} + 2.41535670165126845144 \cdot 10^7 \cdot x^{-6} \\ ~~~ + 1.47478952192985464958 \cdot 10^9 \cdot x^{-8} + 4.58595115847765779830 \cdot 10^{10} \cdot x^{-10} + 7.08501308149515401563 \cdot 10^{11} \cdot x^{-12} \\ ~~~ + 5.06084464593475076774 \cdot 10^{12} \cdot x^{-14} + 1.43468549171581016479 \cdot 10^{13} \cdot x^{-16} + 1.11535493509914254097 \cdot 10^{13} \cdot x^{-18} \end{array} } \right) \\ & &\\ g(x) &\approx & \dfrac{1}{x^2} \cdot \left(\frac{ \begin{array}{l} 1 + 8.1359520115168615 \cdot 10^2 \cdot x^{-2} + 2.35239181626478200 \cdot 10^5 \cdot x^{-4} +3.12557570795778731 \cdot 10^7 \cdot x^{-6} \\ ~~~ + 2.06297595146763354 \cdot 10^9 \cdot x^{-8} + 6.83052205423625007 \cdot 10^{10} \cdot x^{-10} + 1.09049528450362786 \cdot 10^{12} \cdot x^{-12} \\ ~~~ + 7.57664583257834349 \cdot 10^{12} \cdot x^{-14} + 1.81004487464664575 \cdot 10^{13} \cdot x^{-16} + 6.43291613143049485 \cdot 10^{12} \cdot x^{-18} \\ ~~~ - 1.36517137670871689 \cdot 10^{12} \cdot x^{-20} \end{array} }{ \begin{array}{l} 1 + 8.19595201151451564 \cdot 10^2 \cdot x^{-2} + 2.40036752835578777 \cdot 10^5 \cdot x^{-4} + 3.26026661647090822 \cdot 10^7 \cdot x^{-6} \\ ~~~ + 2.23355543278099360 \cdot 10^9 \cdot x^{-8} + 7.87465017341829930 \cdot 10^{10} \cdot x^{-10} + 1.39866710696414565 \cdot 10^{12} \cdot x^{-12} \\ ~~~ + 1.17164723371736605 \cdot 10^{13} \cdot x^{-14} + 4.01839087307656620 \cdot 10^{13} \cdot x^{-16} + 3.99653257887490811 \cdot 10^{13} \cdot x^{-18} \end{array} } \right) \\ \end{array}</math>
See also
References
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