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The special function defined for real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051650/i0516501.png" /> by
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The special function defined for real $x$ by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051650/i0516502.png" /></td> </tr></table>
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$$\operatorname{Si}(x)=\int\limits_0^x\frac{\sin t}{t}\,dt.$$
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051650/i0516503.png" /> one has
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For $x>0$ one has
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051650/i0516504.png" /></td> </tr></table>
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$$\operatorname{Si}(x)=\frac\pi2-\int\limits_x^\infty\frac{\sin t}{t}\,dt.$$
  
 
One sometimes uses the notation
 
One sometimes uses the notation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051650/i0516505.png" /></td> </tr></table>
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$$\operatorname{si}(x)=-\int\limits_x^\infty\frac{\sin t}{t}\,dt\equiv\operatorname{Si}(x)-\frac\pi2.$$
  
 
Some particular values are:
 
Some particular values are:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051650/i0516506.png" /></td> </tr></table>
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$$\operatorname{Si}(0)=0,\qquad\operatorname{Si}(\infty)=\frac\pi2,\qquad\operatorname{si}(\infty)=0.$$
  
 
Some special relations:
 
Some special relations:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051650/i0516507.png" /></td> </tr></table>
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$$\operatorname{Si}(-x)=-\operatorname{Si}(x);\qquad\operatorname{si}(x)+\operatorname{si}(-x)=-\pi;$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051650/i0516508.png" /></td> </tr></table>
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$$\int\limits_0^\infty\operatorname{si}^2(t)\,dt=\frac\pi2;\qquad\int\limits_0^\infty e^{-pt}\operatorname{si}(qt)\,dt=-\frac1p\arctan\frac pq;$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051650/i0516509.png" /></td> </tr></table>
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$$\int\limits_0^\infty\sin t\operatorname{si}(t)\,dt=-\frac\pi4;\qquad\int\limits_0^\infty\operatorname{Ci}(t)\operatorname{si}(t)\,dt=-\ln2,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051650/i05165010.png" /> is the [[Integral cosine|integral cosine]]. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051650/i05165011.png" /> small,
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where $\operatorname{Ci}(t)$ is the [[Integral cosine|integral cosine]]. For $x$ small,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051650/i05165012.png" /></td> </tr></table>
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$$\operatorname{Si}(x)\approx x.$$
  
The asymptotic representation for large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051650/i05165013.png" /> is
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The asymptotic representation for large $x$ is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051650/i05165014.png" /></td> </tr></table>
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$$\operatorname{Si}(x)=\frac\pi2-\frac{\cos x}{x}P(x)-\frac{\sin x}{x}Q(x),$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051650/i05165015.png" /></td> </tr></table>
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$$P(x)\sim\sum_{k=0}^\infty\frac{(-1)^k(2k)!}{x^{2k}},$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051650/i05165016.png" /></td> </tr></table>
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$$Q(x)\sim\sum_{k=0}^\infty\frac{(-1)^k(2k+1)!}{x^{2k+1}}.$$
  
 
The integral sine has the series representation
 
The integral sine has the series representation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051650/i05165017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$\operatorname{Si}(x)=x-\frac{x^3}{3!\,3}+\dotsb+(-1)^k\frac{x^{2k+1}}{(2k+1)!\,(2k+1)}+\dotsb.\label{*}\tag{*}$$
  
As a function of the complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051650/i05165018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051650/i05165019.png" />, defined by (*), is an entire function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051650/i05165020.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051650/i05165021.png" />-plane.
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As a function of the complex variable $z$, $\operatorname{Si}(z)$, defined by \eqref{*}, is an entire function of $z$ in the $z$-plane.
  
The integral sine is related to the [[Integral exponential function|integral exponential function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051650/i05165022.png" /> by
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The integral sine is related to the [[Integral exponential function|integral exponential function]] $\operatorname{Ei}(z)$ by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051650/i05165023.png" /></td> </tr></table>
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$$\operatorname{si}(z)=\frac{1}{2i}[\operatorname{Ei}(iz)-\operatorname{Ei}(-iz)].$$
  
 
See also [[Si-ci-spiral|Si-ci-spiral]].
 
See also [[Si-ci-spiral|Si-ci-spiral]].
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====Comments====
 
====Comments====
 
This function is better known as the sine integral.
 
This function is better known as the sine integral.
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[[Category:Special functions]]

Latest revision as of 21:27, 1 January 2019

The special function defined for real $x$ by

$$\operatorname{Si}(x)=\int\limits_0^x\frac{\sin t}{t}\,dt.$$

For $x>0$ one has

$$\operatorname{Si}(x)=\frac\pi2-\int\limits_x^\infty\frac{\sin t}{t}\,dt.$$

One sometimes uses the notation

$$\operatorname{si}(x)=-\int\limits_x^\infty\frac{\sin t}{t}\,dt\equiv\operatorname{Si}(x)-\frac\pi2.$$

Some particular values are:

$$\operatorname{Si}(0)=0,\qquad\operatorname{Si}(\infty)=\frac\pi2,\qquad\operatorname{si}(\infty)=0.$$

Some special relations:

$$\operatorname{Si}(-x)=-\operatorname{Si}(x);\qquad\operatorname{si}(x)+\operatorname{si}(-x)=-\pi;$$

$$\int\limits_0^\infty\operatorname{si}^2(t)\,dt=\frac\pi2;\qquad\int\limits_0^\infty e^{-pt}\operatorname{si}(qt)\,dt=-\frac1p\arctan\frac pq;$$

$$\int\limits_0^\infty\sin t\operatorname{si}(t)\,dt=-\frac\pi4;\qquad\int\limits_0^\infty\operatorname{Ci}(t)\operatorname{si}(t)\,dt=-\ln2,$$

where $\operatorname{Ci}(t)$ is the integral cosine. For $x$ small,

$$\operatorname{Si}(x)\approx x.$$

The asymptotic representation for large $x$ is

$$\operatorname{Si}(x)=\frac\pi2-\frac{\cos x}{x}P(x)-\frac{\sin x}{x}Q(x),$$

where

$$P(x)\sim\sum_{k=0}^\infty\frac{(-1)^k(2k)!}{x^{2k}},$$

$$Q(x)\sim\sum_{k=0}^\infty\frac{(-1)^k(2k+1)!}{x^{2k+1}}.$$

The integral sine has the series representation

$$\operatorname{Si}(x)=x-\frac{x^3}{3!\,3}+\dotsb+(-1)^k\frac{x^{2k+1}}{(2k+1)!\,(2k+1)}+\dotsb.\label{*}\tag{*}$$

As a function of the complex variable $z$, $\operatorname{Si}(z)$, defined by \eqref{*}, is an entire function of $z$ in the $z$-plane.

The integral sine is related to the integral exponential function $\operatorname{Ei}(z)$ by

$$\operatorname{si}(z)=\frac{1}{2i}[\operatorname{Ei}(iz)-\operatorname{Ei}(-iz)].$$

See also Si-ci-spiral.

For references, and the graph of the integral sine, see Integral cosine.


Comments

This function is better known as the sine integral.

How to Cite This Entry:
Integral sine. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integral_sine&oldid=11622
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article