trigonometric integral#Cosine integral

Image:Sine integral.svg

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

The different sine integral definitions are

$$\backslash operatorname\{Si\}(x)\; =\; \backslash int\_0^x\backslash frac\{\backslash sin\; t\}\{t\}\backslash ,dt$$

$$\backslash operatorname\{si\}(x)\; =\; -\backslash int\_x^\backslash infty\backslash frac\{\backslash sin\; t\}\{t\}\backslash ,dt~.$$

Note that the integrand $\backslash frac\{\backslash sin(t)\}\{t\}$ is the sinc function, and also the zeroth spherical Bessel function.

Since {{math|sinc}} is an even entire function (holomorphic over the entire complex plane), {{math|Si}} is entire, odd, and the integral in its definition can be taken along 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,

$$\backslash operatorname\{Si\}(x)\; -\; \backslash operatorname\{si\}(x)\; =\; \backslash int\_0^\backslash infty\backslash frac\{\backslash sin\; t\}\{t\}\backslash ,dt\; =\; \backslash frac\{\backslash pi\}\{2\}\; \backslash quad\; \backslash text\{\; or\; \}\; \backslash quad\; \backslash operatorname\{Si\}(x)\; =\; \backslash frac\{\backslash pi\}\{2\}\; +\; \backslash operatorname\{si\}(x)\; ~.$$

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.

Image:Cosine integral.svg

The different cosine integral definitions are

$$\backslash operatorname\{Cin\}(x)\; ~\backslash equiv~\; \backslash int\_0^x\; \backslash frac\{\backslash \; 1\; -\; \backslash cos\; t\backslash \; \}\{\; t\; \}\backslash \; \backslash operatorname\{d\}\; t\; ~.$$

{{math|Cin}} is an even, entire function. For that reason, some texts define {{math|Cin}} as the primary function, and derive {{math|Ci}} in terms of {{math|Cin .}}

$$\backslash operatorname\{Ci\}(x)\; ~~\backslash equiv~\; -\backslash int\_x^\backslash infty\; \backslash frac\{\backslash \; \backslash cos\; t\backslash \; \}\{\; t\; \}\backslash \; \backslash operatorname\{d\}\; t\; ~$$

$$~~\; \backslash qquad\; ~=~~\; \backslash gamma\; ~+~\; \backslash ln\; x\; ~-~\; \backslash int\_0^x\; \backslash frac\{\backslash \; 1\; -\; \backslash cos\; t\backslash \; \}\{\; t\; \}\backslash \; \backslash operatorname\{d\}\; t\; ~$$

$$~~\; \backslash qquad\; ~=~~\; \backslash gamma\; ~+~\; \backslash ln\; x\; ~-~\; \backslash operatorname\{Cin\}\; x\; ~$$

for where {{math|γ ≈ 0.57721566490 ...}} 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 plot above) that arises because of a branch cut in the standard logarithm function ({{math|ln}}).

{{math|Ci(x)}} is the antiderivative of {{math|{{sfrac|cos x| x }} }} (which vanishes as $\backslash \; x\; \backslash to\; \backslash infty\backslash$). The two definitions are related by

$$\backslash operatorname\{Ci\}(x)\; =\; \backslash gamma\; +\; \backslash ln\; x\; -\; \backslash operatorname\{Cin\}(x)\; ~.$$

The hyperbolic sine integral is defined as

$$\backslash operatorname\{Shi\}(x)\; =\backslash int\_0^x\; \backslash frac\; \{\backslash sinh\; (t)\}\{t\}\backslash ,dt.$$

It is related to the ordinary sine integral by

$$\backslash operatorname\{Si\}(ix)\; =\; i\backslash operatorname\{Shi\}(x).$$

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

where $\backslash gamma$ is the Euler–Mascheroni constant.

It has the series expansion

$$\backslash operatorname\{Chi\}(x)\; =\; \backslash gamma\; +\; \backslash ln(x)\; +\; \backslash frac\; \{x^2\}\{4\}\; +\; \backslash frac\; \{x^4\}\{96\}\; +\; \backslash frac\; \{x^6\}\{4320\}\; +\; \backslash frac\; \{x^8\}\{322560\}\; +\; \backslash frac\{x^\{10\}\}\{36288000\}\; +\; O(x^\{12\}).$$

Trigonometric integrals can be understood in terms of the so-called "auxiliary functions"

\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}

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])

$$\backslash begin\{array\}\{rcl\}$$

\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}

Image:Nielsen's spiral.png

The spiral formed by parametric plot of {{math|si, ci}} is known as Nielsen's spiral.

$$x(t)\; =\; a\; \backslash times\; \backslash operatorname\{ci\}(t)$$

$$y(t)\; =\; a\; \backslash times\; \backslash operatorname\{si\}(t)$$

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.{{cite book|last=Gray|title=Modern Differential Geometry of Curves and Surfaces.|publisher=|year=1993|isbn=|location=Boca Raton|pages=119}}

Various expansions can be used for evaluation of trigonometric integrals, depending on the range of the argument.

$$\backslash operatorname\{Si\}(x)\; \backslash sim\; \backslash frac\{\backslash pi\}\{2\}$$

- \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)

$$\backslash operatorname\{Ci\}(x)\; \backslash sim\; \backslash frac\{\backslash sin\; x\}\{x\}\backslash left(1-\backslash frac\{2!\}\{x^2\}+\backslash frac\{4!\}\{x^4\}-\backslash frac\{6!\}\{x^6\}\backslash cdots\backslash right)$$

- \frac{\cos x}{x}\left(\frac{1}{x}-\frac{3!}{x^{3}}+\frac{5!}{x^5}-\frac{7!}{x^7}\cdots\right) ~.

These series are asymptotic and divergent, although can be used for estimates and even precise evaluation at {{math|ℜ(x) ≫ 1}}.

$$\backslash operatorname\{Si\}(x)=\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{(-1)^\{n\}x^\{2n+1\}\}\{(2n+1)(2n+1)!\}=x-\backslash frac\{x^3\}\{3!\backslash cdot3\}+\backslash frac\{x^5\}\{5!\backslash cdot5\}-\backslash frac\{x^7\}\{7!\; \backslash cdot7\}\backslash pm\backslash cdots$$

$$\backslash operatorname\{Ci\}(x)=\; \backslash gamma+\backslash ln\; x+\backslash sum\_\{n=1\}^\{\backslash infty\}\backslash frac\{(-1)^\{n\}x^\{2n\}\}\{2n(2n)!\}=\backslash gamma+\backslash ln\; x-\backslash frac\{x^2\}\{2!\backslash cdot2\}\; +\; \backslash frac\{x^4\}\{4!\; \backslash cdot4\}\backslash mp\backslash cdots$$

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.

From the Maclaurin series expansion of sine:

$$\backslash sin\backslash ,x\; =\; x\; -\; \backslash frac\{x^3\}\{3!\}+\backslash frac\{x^5\}\{5!\}-\; \backslash frac\{x^7\}\{7!\}+\backslash frac\{x^9\}\{9!\}-\backslash frac\{x^\{11\}\}\{11!\}\; +\; \backslash cdots$$

$$\backslash frac\{\backslash sin\backslash ,x\}\{x\}\; =\; 1\; -\; \backslash frac\{x^2\}\{3!\}+\backslash frac\{x^4\}\{5!\}-\; \backslash frac\{x^6\}\{7!\}+\backslash frac\{x^8\}\{9!\}-\backslash frac\{x^\{10\}\}\{11!\}+\backslash cdots$$

$$\backslash therefore\backslash int\; \backslash frac\{\backslash sin\backslash ,x\}\{x\}dx\; =\; x\; -\; \backslash frac\{x^3\}\{3!\backslash cdot3\}+\backslash frac\{x^5\}\{5!\backslash cdot5\}-\; \backslash frac\{x^7\}\{7!\backslash cdot7\}+\backslash frac\{x^9\}\{9!\backslash cdot9\}-\backslash frac\{x^\{11\}\}\{11!\backslash cdot11\}+\backslash cdots$$

The function

$$\backslash operatorname\{E\}\_1(z)\; =\; \backslash int\_1^\backslash infty\; \backslash frac\{\backslash exp(-zt)\}\{t\}\backslash ,dt\; \backslash qquad~\backslash text\{\; for\; \}~\; \backslash Re(z)\; \backslash ge\; 0$$

is called the exponential integral. It is closely related to {{math|Si}} and {{math|Ci}},

\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 ~.

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

\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} ~,

which is the real part of

\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} ~.

Similarly

\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}~.

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),{{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 }} are accurate to better than {{math|10⁻¹⁶}} for {{math|0 ≤ x ≤ 4}},

$$\backslash begin\{array\}\{rcl\}$$

\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}

The integrals may be evaluated indirectly via auxiliary functions $f(x)$ and $g(x)$, which are defined by

For $x\; \backslash ge\; 4$ the Padé rational functions given below approximate $f(x)$ and $g(x)$ with error less than 10⁻¹⁶:

$$\backslash begin\{array\}\{rcl\}$$

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}

• Logarithmic integral
• Tanc function
• Tanhc function
• Sinhc function
• Coshc function

{{reflist}}

{{refbegin}}

• {{AS ref|5|231}}

{{refend}}

{{refbegin}}

• {{cite arXiv |first1=R.J. |last1=Mathar |eprint=0912.3844 |title=Numerical evaluation of the oscillatory integral over exp(iπx)·x^1/x 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 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}}

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

{{Nonelementary Integral}}

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