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| In [[mathematics]], the '''trigonometric integrals''' are a [[indexed family|family]] of [[integral]]s which involve [[trigonometric function]]s. A number of the basic trigonometric integrals are discussed at the [[list of integrals of trigonometric functions]].
| |
| | |
| ==Sine integral==
| |
| [[Image:Sine integral.svg|thumb|right|Plot of '''Si(''x'')''' for 0 ≤ ''x'' ≤ 8π.]]
| |
| The different [[sine]] integral definitions are:
| |
| | |
| :<math>{\rm Si}(x) = \int_0^x\frac{\sin t}{t}\,dt</math> | |
| | |
| :<math>{\rm si}(x) = -\int_x^\infty\frac{\sin t}{t}\,dt</math>
| |
| | |
| So by definition, <math>{\rm Si}(x)</math> is the [[primitive]] of <math>\sin x/x</math> which is zero for <math>x=0</math> and <math>{\rm si}(x)</math> is the primitive of <math>\sin x/x</math> which is zero for <math>x=\infty</math>. The relation is given by
| |
| | |
| :<math>{\rm Si}(x) - {\rm si}(x) = \int_0^\infty\frac{\sin t}{t}\,dt = \frac{\pi}{2},</math>
| |
| | |
| where the last integral is known as the [[Dirichlet integral]]. Note that <math>\frac{\sin t}{t}</math> is the [[sinc function]] and also the zeroth [[Bessel function#Spherical Bessel functions: jn.2C yn|spherical Bessel function]].
| |
| | |
| In [[signal processing]], the oscillations of the Sine integral cause [[overshoot (signal)|overshoot]] and [[ringing artifacts]] when using the [[sinc filter]], and [[frequency domain]] ringing if using a truncated sinc filter as a [[low-pass filter]].
| |
| | |
| The [[Gibbs phenomenon]] is a related phenomenon: thinking of sinc as a [[low-pass filter]] and the Sine integral as its [[convolution]] with the [[Heaviside step function]], it corresponds to truncating the [[Fourier series]], which causes the Gibbs phenomenon.
| |
| | |
| ==Cosine integral==
| |
| [[Image:Cosine integral.svg|thumb|right|Plot of '''Ci(''x'')''' for 0 < ''x'' ≤ 8π.]]
| |
| The different [[cosine]] integral definitions are:
| |
| | |
| :<math>{\rm Ci}(x) = \gamma + \ln x + \int_0^x\frac{\cos t-1}{t}\,dt</math>
| |
| | |
| :<math>{\rm ci}(x) = -\int_x^\infty\frac{\cos t}{t}\,dt</math>
| |
| | |
| :<math>{\rm Cin}(x) = \int_0^x\frac{1-\cos t}{t}\,dt</math>
| |
| | |
| where <math>\gamma</math> is the [[Euler–Mascheroni constant]].
| |
| | |
| <math>{\rm ci}(x)</math> is the primitive of <math>\cos x/x</math> which is zero for <math>x=\infty</math>. We have: | |
| | |
| :<math>{\rm ci}(x)={\rm Ci}(x)\,</math>
| |
| :<math>{\rm Cin}(x)=\gamma+\ln x-{\rm Ci}(x)\,</math>
| |
| | |
| ==Hyperbolic sine integral==
| |
| The [[hyperbolic sine]] integral:
| |
| | |
| :<math>{\rm Shi}(x) = \int_0^x\frac{\sinh t}{t}\,dt = {\rm shi}(x).</math>
| |
| :<math>{\rm Shi}(x)=\sum_{n=0}^\infty \frac{x^{2n+1}}{(2n+1)^2(2n)!}=x+\frac{x^3}{3!\cdot3}+\frac{x^5}{5!\cdot5}+\frac{x^7}{7! \cdot7}+\cdots.</math>
| |
| | |
| ==Hyperbolic cosine integral==
| |
| The [[hyperbolic cosine]] integral is
| |
| | |
| :<math>{\rm Chi}(x) = \gamma+\ln x + \int_0^x\frac{\cosh t-1}{t}\,dt = {\rm chi}(x)</math>
| |
| | |
| ==Nielsen's spiral==
| |
| [[Image:Nielsen's spiral.png|thumb|right|Nielsen's spiral.]]
| |
| The [[spiral]] formed by parametric plot of si,ci is known as [[Nielsen's spiral]]. It is also referred to as the [[Euler spiral]], [http://mathworld.wolfram.com/CornuSpiral.html the Cornu spiral], a clothoid, or as a linear-curvature polynomial spiral. The spiral is also closely related to the [[Fresnel integral]]s. This spiral has applications in vision processing, road and track construction and other areas. | |
| | |
| ==Expansion==
| |
| Various expansions can be used for evaluation of Trigonometric integrals, depending on the range of the argument.
| |
| | |
| ===Asymptotic series (for large argument)===
| |
| :<math>{\rm Si}(x)=\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>{\rm Ci}(x)= \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 series|asymptotic]] and divergent, although can be used for estimates and even precise evaluation at <math>~{\rm Re} (x) \gg 1~</math>. | |
| | |
| ===Convergent series===
| |
| :<math>{\rm 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>{\rm 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 <math>~x~</math>, although for <math>|x|\gg 1</math> the series will converge slowly initially, requiring many terms for high precisions.
| |
| | |
| ==Relation with the exponential integral of imaginary argument==
| |
| The function
| |
| | |
| : <math> {\rm E}_1(z) = \int_1^\infty \frac{\exp(-zt)}{t}\,{\rm d} t \qquad({\rm Re}(z) \ge 0) </math>
| |
| | |
| is called the [[exponential integral]]. It is closely related to Si and Ci:
| |
| | |
| :<math>
| |
| {\rm E}_1( {\rm i}\!~ x) = i\left(-\frac{\pi}{2} + {\rm Si}(x)\right)-{\rm Ci}(x) = i~{\rm si}(x) - {\rm ci}(x) \qquad(x>0)
| |
| </math> | |
| | |
| As each involved function is analytic except the cut at negative values of the argument,
| |
| the area of validity of the relation should be extended to <math>{\rm Re}(x)>0</math>.
| |
| (Out of this range, additional terms which are integer factors of <math>\pi</math> appear in the expression).
| |
| | |
| Cases of imaginary argument of the generalized integro-exponential function are
| |
| | |
| : <math>
| |
| \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>
| |
| \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(\gamma+\ln a) + \sum_{n\ge 1}\frac{(ia)^n}{n!n^2}.
| |
| </math> | |
| | |
| Similarly
| |
| | |
| : <math>
| |
| \int_1^\infty e^{iax}\frac{\ln x}{x^2}dx
| |
| =1+ia[-\frac{\pi^2}{24}+\gamma\left(\frac{\gamma}{2}+\ln a-1\right)+\frac{\ln^2 a}{2}-\ln a+1
| |
| -\frac{i\pi}{2}(\gamma+\ln a-1)]+\sum_{n\ge 1}\frac{(ia)^{n+1}}{(n+1)!n^2}.
| |
| </math>
| |
| | |
| ==Efficient evaluation==
| |
| | |
| [[Padé_approximant|Padé approximants]] of the convergent Taylor series provide an efficient way to evaluate the functions for small arguments. The following formulae are accurate to better than <math>10^{-16}</math> for <math>0 \le x \le 4</math>:
| |
| | |
| <br> | |
| <math>
| |
| \begin{array}{rcl}
| |
| {\rm Si}(x) &=& 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)\\
| |
| &~&\\
| |
| {\rm Ci}(x) &=& \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>
| |
| | |
| <br> | |
| | |
| For <math>x > 4</math>, one can use the helper functions:
| |
| | |
| <br>
| |
| <math>
| |
| \begin{array}{rcl}
| |
| f(x)
| |
| &=& \int_0^\infty \frac{sin(t)}{t+x} dt = \int_0^\infty \frac{e^{-x t}}{t^2 + 1} dt
| |
| ~=~ {\rm Ci}(x) \sin(x) + \left(\frac{\pi}{2} - {\rm Si}(x) \right) \cos(x) \\
| |
| g(x)
| |
| &=& \int_0^\infty \frac{cos(t)}{t+x} dt = \int_0^\infty \frac{t e^{-x t}}{t^2 + 1} dt
| |
| ~=~ -{\rm Ci}(x) \cos(x) + \left(\frac{\pi}{2} - {\rm Si}(x) \right) \sin(x) \\
| |
| \end{array}
| |
| </math>
| |
| | |
| <br>
| |
| using which, the trigonometric integrals may be expressed as
| |
| | |
| <br>
| |
| <math>
| |
| \begin{array}{rcl}
| |
| {\rm Si}(x) &=& \frac{\pi}{2} - f(x) \cos(x) - g(x) \sin(x) \\
| |
| {\rm Ci}(x) &=& f(x) \sin(x) - g(x) \cos(x) \\
| |
| \end{array}
| |
| </math>
| |
| | |
| <br>
| |
| Chebyshev-Padé expansions of <math>\;\;\frac{1}{\sqrt{y}} \; f\left(\frac{1}{\sqrt{y}} \right) \;\;</math> and <math>\;\;\frac{1}{y} \; g\left(\frac{1}{\sqrt{y}} \right)\;\; </math>
| |
| in the interval <math>0..\frac{1}{4^2}</math> give the following approximants, good to better than <math>10^{-16}</math> for <math>x \ge 4</math>:
| |
| | |
| <br>
| |
| <math>
| |
| \begin{array}{rcl}
| |
| f(x) &=& \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) &=& \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>
| |
| | |
| | |
| <br>
| |
| Here are text versions of the above suitable for copying into computer code (using x2 = x*x and y = 1/(x*x) where appropriate):
| |
| | |
| Si = x*(1. +
| |
| x2*(-4.54393409816329991e-2 +
| |
| x2*(1.15457225751016682e-3 +
| |
| x2*(-1.41018536821330254e-5 +
| |
| x2*(9.43280809438713025e-8 +
| |
| x2*(-3.53201978997168357e-10 +
| |
| x2*(7.08240282274875911e-13 +
| |
| x2*(-6.05338212010422477e-16))))))))
| |
| / (1. +
| |
| x2*(1.01162145739225565e-2 +
| |
| x2*(4.99175116169755106e-5 +
| |
| x2*(1.55654986308745614e-7 +
| |
| x2*(3.28067571055789734e-10 +
| |
| x2*(4.5049097575386581e-13 +
| |
| x2*(3.21107051193712168e-16)))))))
| |
|
| |
| Ci = 0.577215664901532861 + ln(x) +
| |
| x2*(-0.25 +
| |
| x2*(7.51851524438898291e-3 +
| |
| x2*(-1.27528342240267686e-4 +
| |
| x2*(1.05297363846239184e-6 +
| |
| x2*(-4.68889508144848019e-9 +
| |
| x2*(1.06480802891189243e-11 +
| |
| x2*(-9.93728488857585407e-15)))))))
| |
| / (1. +
| |
| x2*(1.1592605689110735e-2 +
| |
| x2*(6.72126800814254432e-5 +
| |
| x2*(2.55533277086129636e-7 +
| |
| x2*(6.97071295760958946e-10 +
| |
| x2*(1.38536352772778619e-12 +
| |
| x2*(1.89106054713059759e-15 +
| |
| x2*(1.39759616731376855e-18))))))))
| |
|
| |
| f = (1. +
| |
| y*(7.44437068161936700618e2 +
| |
| y*(1.96396372895146869801e5 +
| |
| y*(2.37750310125431834034e7 +
| |
| y*(1.43073403821274636888e9 +
| |
| y*(4.33736238870432522765e10 +
| |
| y*(6.40533830574022022911e11 +
| |
| y*(4.20968180571076940208e12 +
| |
| y*(1.00795182980368574617e13 +
| |
| y*(4.94816688199951963482e12 +
| |
| y*(-4.94701168645415959931e11)))))))))))
| |
| / (x*(1. +
| |
| y*(7.46437068161927678031e2 +
| |
| y*(1.97865247031583951450e5 +
| |
| y*(2.41535670165126845144e7 +
| |
| y*(1.47478952192985464958e9 +
| |
| y*(4.58595115847765779830e10 +
| |
| y*(7.08501308149515401563e11 +
| |
| y*(5.06084464593475076774e12 +
| |
| y*(1.43468549171581016479e13 +
| |
| y*(1.11535493509914254097e13)))))))))))
| |
|
| |
| g = y*(1. +
| |
| y*(8.1359520115168615e2 +
| |
| y*(2.35239181626478200e5 +
| |
| y*(3.12557570795778731e7 +
| |
| y*(2.06297595146763354e9 +
| |
| y*(6.83052205423625007e10 +
| |
| y*(1.09049528450362786e12 +
| |
| y*(7.57664583257834349e12 +
| |
| y*(1.81004487464664575e13 +
| |
| y*(6.43291613143049485e12 +
| |
| y*(-1.36517137670871689e12)))))))))))
| |
| / (1. +
| |
| y*(8.19595201151451564e2 +
| |
| y*(2.40036752835578777e5 +
| |
| y*(3.26026661647090822e7 +
| |
| y*(2.23355543278099360e9 +
| |
| y*(7.87465017341829930e10 +
| |
| y*(1.39866710696414565e12 +
| |
| y*(1.17164723371736605e13 +
| |
| y*(4.01839087307656620e13 +
| |
| y*(3.99653257887490811e13))))))))))
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| | |
| ==See also==
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| * [[Exponential integral]]
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| * [[Logarithmic integral]]
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| | |
| === Signal processing ===
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| * [[Gibbs phenomenon]]
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| * [[Ringing artifacts]]
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| | |
| == References ==
| |
| {{Reflist}}
| |
| *{{AS ref|5|231}}
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| *{{Citation | last1=Press | first1=WH | last2=Teukolsky | first2=SA | last3=Vetterling | first3=WT | last4=Flannery | first4=BP | year=2007 | title=Numerical Recipes: The Art of Scientific Computing | edition=3rd | publisher=Cambridge University Press | publication-place=New York | isbn=978-0-521-88068-8 | chapter=Section 6.8.2. Cosine and Sine Integrals | chapter-url=http://apps.nrbook.com/empanel/index.html#pg=300}}
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| *{{dlmf|id=6|title=Exponential, Logarithmic, Sine, and Cosine Integrals|first=N. M. |last=Temme}}
| |
| * {{ cite arXiv|
| |
| |first1=R. J.
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| |last1=Mathar
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| |eprint=0912.3844
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| |title=Numerical evaluation of the oscillatory integral over exp(''i{{pi}}x'')·''x''<sup>1/''x''</sup> between 1 and ∞
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| |year=2009
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| }}, Appendix B.
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| *[http://de2de.synechism.org/c5/sec58.pdf Sine Integral Taylor series proof.]
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| | |
| ==External links==
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| * http://mathworld.wolfram.com/SineIntegral.html
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| * {{springer|title=Integral sine|id=p/i051650}}
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| * {{springer|title=Integral cosine|id=p/i051370}}
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| {{DEFAULTSORT:Trigonometric Integral}}
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| [[Category:Trigonometry]]
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| [[Category:Special functions]]
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| [[Category:Special hypergeometric functions]]
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| [[Category:Integrals]]
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| [[ru:Интегральные тригонометрические функции]]
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