Editing Trigonometric integral

{{for|simple integrals of trigonometric functions|List of integrals of trigonometric functions}}
[[File:Plot of the hyperbolic sine integral function Shi(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg|alt=Plot of the hyperbolic sine integral function Shi(''z'') in the complex plane from −2&nbsp;−&nbsp;2''i'' to 2&nbsp;+&nbsp;2''i''|thumb|Plot of the hyperbolic sine integral function {{math|Shi(''z'')}} in the complex plane from {{math|−2 − 2''i''}} to {{math|2 + 2''i''}}]]
{{Use American English|date = January 2019}}
{{Short description|Special function defined by an integral}}
[[Image:sine cosine integral.svg|thumb|{{math|Si(''x'')}} (blue) and {{math|Ci(''x'')}} (green) shown on the same plot.]]

[[File:Integral sine in the complex plain.svg|thumb|Sine integral in the complex plane, plotted with a variant of [[domain coloring]].]]

[[File:Cosc.svg|thumb|{{anchor|ci_plot_anchor}}Cosine integral in the complex plane. Note the [[branch cut]] along the negative real axis.]]

In [[mathematics]], '''trigonometric integrals''' are a [[indexed family|family]] of [[nonelementary integral]]s involving [[trigonometric function]]s.

== Sine integral ==
[[Image:Sine integral.svg|thumb|Plot of {{math|Si(''x'')}} for {{math|0 ≤ ''x'' ≤ 8''π''}}.]]
[[File:Plot of the cosine integral function Ci(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg|alt=Plot of the cosine integral function Ci(''z'') in the complex plane from −2&nbsp;−&nbsp;2''i'' to 2&nbsp;+&nbsp;2''i''|thumb|Plot of the cosine integral function {{math|Ci(''z'')}} in the complex plane from {{math|−2 − 2''i''}} to {{math|2 + 2''i''}}]]
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 [[Bessel function#Spherical Bessel functions: jn.2C yn|spherical Bessel function]].
Since {{math|sinc}} is an [[even function|even]] [[entire function]] ([[holomorphic]] over the entire complex plane), {{math|Si}} is entire, odd, and the integral in its definition can be taken along [[Cauchy's integral theorem|any path]] connecting the endpoints.

By definition, {{math|Si(''x'')}} is the [[antiderivative]] of {{math|sin ''x'' / ''x''}} whose value is zero at {{math|1=''x'' = 0}}, and {{math|si(''x'')}} is the antiderivative whose value is zero at {{math|1=''x'' = ∞}}. 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 (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]].

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 ==
[[Image:Cosine integral.svg|thumb|Plot of {{math|Ci(''x'')}} for {{math|0 < ''x'' ≤ 8''π''}}]]
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>

{{math|Cin}} is an [[even and odd functions|even]], [[entire function]]. For that reason, some texts define {{math|Cin}} as the primary function, and derive {{math|Ci}} in terms of {{math|Cin&nbsp;.}}

<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 {{math|''γ'' ≈ 0.57721566490&nbsp;...}} is the [[Euler–Mascheroni constant]]. Some texts use {{math|ci}} instead of {{math|Ci}}. The restriction on {{math|Arg(x)}} is to avoid a discontinuity (shown as the orange vs blue area on the left half of the [[#ci_plot_anchor|plot above]]) that arises because of a [[branch cut]] in the standard [[natural logarithm|logarithm function]] ({{math|ln}}).

{{math|Ci(''x'')}} is the antiderivative of {{math|{{sfrac|cos ''x''| ''x'' }} }} (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
[[File:Plot of the hyperbolic cosine integral function Chi(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg|alt=Plot of the hyperbolic cosine integral function Chi(''z'') in the complex plane from −2&nbsp;−&nbsp;2''i'' to 2&nbsp;+&nbsp;2''i''|thumb|Plot of the hyperbolic cosine integral function {{math|Chi(''z'')}} in the complex plane from {{math|−2 − 2''i''}} to {{math|2 + 2''i''}}]]
<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 function|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, [http://people.math.sfu.ca/~cbm/aands/page_232.htm 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 ==
[[Image:Nielsen's spiral.png|thumb|Nielsen's spiral.]]
The [[spiral]] formed by parametric plot of {{math|si, ci}} 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 integral]]s and the [[Euler spiral]]. Nielsen's spiral has applications in vision processing, road and track construction and other areas.<ref>{{cite book|last=Gray|title=Modern Differential Geometry of Curves and Surfaces.|publisher=|year=1993|isbn=|location=Boca Raton|pages=119}}</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 series|asymptotic]] and divergent, although can be used for estimates and even precise evaluation at {{math|ℜ(''x'') ≫ 1}}.

=== 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 {{mvar|x}}, although for {{math|{{abs|''x''}} ≫ 1}}, 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 {{math|Si}} and {{math|Ci}},
<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 {{math|''π''}} 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 approximant]]s 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>{{cite journal |last1=Rowe |first1=B. |display-authors=etal |title=GALSIM: The modular galaxy image simulation toolkit |journal=Astronomy and Computing |year=2015 |volume=10 |page=121 |doi=10.1016/j.ascom.2015.02.002 |arxiv=1407.7676 |bibcode=2015A&C....10..121R|s2cid=62709903 }}</ref> are accurate to better than {{math|10<sup>−16</sup>}} for {{math|0 ≤ ''x'' ≤ 4}},
<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 function|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>
|-
|colspan="3" align="center"| <small>''or equivalently''</small> 
|-
|<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 approximant|Padé rational functions]] given below approximate <math>f(x)</math> and <math>g(x)</math> with error less than 10<sup>−16</sup>:<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 ==
* [[Logarithmic integral]]
* [[Tanc function]]
* [[Tanhc function]]
* [[Sinhc function]]
* [[Coshc function]]

== References ==
{{reflist}}
{{refbegin}}
* {{AS ref|5|231}}
{{refend}}

== Further reading ==
{{refbegin}}
* {{cite arXiv |first1=R.J. |last1=Mathar |eprint=0912.3844 |title=Numerical evaluation of the oscillatory integral over exp(''iπx'')·''x''<sup>1/''x''</sup> between 1 and ∞ |year=2009 |class=math.CA |at=Appendix B}}
* {{cite book |last1=Press |first1=W.H. |last2=Teukolsky |first2=S.A. |last3=Vetterling |first3=W.T. |last4=Flannery |first4=B.P. |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&nbsp;6.8.2 – Cosine and Sine Integrals |chapter-url=http://apps.nrbook.com/empanel/index.html#pg=300}}
* {{cite web |first=Dan |last=Sloughter |url=http://de2de.synechism.org/c5/sec58.pdf |title=Sine Integral Taylor series proof |website=Difference Equations to Differential Equations}}
* {{dlmf |id=6 |title=Exponential, Logarithmic, Sine, and Cosine Integrals |first=N.M. |last=Temme}}
{{refend}}

== External links ==
* http://mathworld.wolfram.com/SineIntegral.html
* {{springer|title=Integral sine|id=p/i051650}}
* {{springer|title=Integral cosine|id=p/i051370}}

{{Nonelementary Integral}}
{{DEFAULTSORT:Trigonometric Integral}}
[[Category:Trigonometry]]
[[Category:Special functions]]
[[Category:Special hypergeometric functions]]
[[Category:Integrals]]