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Homesine integral

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# sine integral

The function sine integral (in Latin sinus integralis) from $\mathbb{R}$ to $\mathbb{R}$ is defined as

$\displaystyle\mbox{Si }{x}\;:=\;\int_{0}^{x}\frac{\sin t}{t}\,dt=\int_{0}^{x}% \mbox{sinc}(t)\;dt,$ | (1) |

or alternatively as

$\mbox{Si }{x}\,:=\,\int_{0}^{1}\frac{\sin{tx}}{t}\,dt.$ |

It isn’t an elementary function. The equation (1) implies the Taylor series expansion

$\mbox{Si }{z}=z\!-\!\frac{z^{3}}{3\!\cdot\!3!}\!+\!\frac{z^{5}}{5\!\cdot\!5!}% \!-\!\frac{z^{7}}{7\!\cdot\!7!}\!+-\ldots,$ |

which converges for all complex values $z$ and thus defines an entire transcendental function.

$\mbox{Si }{x}$ satisfies the linear third order differential equation

$xf^{{\prime\prime\prime}}(x)\!+\!2f^{{\prime\prime}}(x)\!+\!xf^{{\prime}}(x)=0.$ |

Remark 1. $\lim_{{x\to\infty}}\mbox{Si }{x}=\frac{\pi}{2}$

Remark 2. There is also another “sine integral”

$\mbox{si }{x}\;:=\;\int_{\infty}^{x}\frac{\sin t}{t}\,dt\;=\;\mbox{Si }{x}-% \frac{\pi}{2}$ |

and the corresponding cosine integral

$\mbox{ci }{x}\;:=\;\int_{\infty}^{x}\frac{\cos t}{t}\,dt=\gamma\!+\ln{x}+\!% \int_{0}^{x}\frac{\cos{t}\!-\!1}{t}\,dt$ |

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

Defines:

sine integral, sinus integralis, cosine integral

Related:

SincFunction, SineIntegralInInfinity, LogarithmicIntegral2, CurvatureOfNielsensSpiral, LaplaceTransformOfIntegralSine, FresnelIntegrals, HyperbolicSineIntegral

Synonym:

sinus integralis, Si

Type of Math Object:

Definition

Major Section:

Reference

Parent:

Groups audience:

## Mathematics Subject Classification

30A99*no label found*

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## Comments

## Opera better than Firefox?

Hi all! My computer has about two months refused to show me (with Firefox and Exploder) the PlanetMath entries. Now I downloaded Opera (8.5) and with it I can see all entries perfectly =o)

Jussi

## Re: Opera better than Firefox?

weird :)

the 3 browsers work fine here

f

G -----> H G

p \ /_ ----- ~ f(G)

\ / f ker f

G/ker f

## Re: Opera better than Firefox?

> weird :)

> the 3 browsers work fine here

Yes, of course your computer is in good shape -- mine has been somehow corrupted (Firefox shows no entries here but all entries in PlanetPhysics!! Something to do with the port 8080...)

Regards,

Jussi

## Re: Opera better than Firefox?

I had the same problem at work versus home with firefox and IE. I thought they were blocking 8080, but everything seems to magically work now (1 week later). It may be something else, surely it cannot just be us two...

Ben

## Re: Opera better than Firefox?

Apparently things of spiritual world...