PlanetMath: sine integral

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sine integral

(Definition)

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

$\displaystyle {\mathrm{Si}}{x} := \int_0^x\frac{\sin t}{t} dt = \int_0^x{\mathrm{sinc}}(t) dt,$

(1)

or alternatively as

$\displaystyle {\mathrm{Si}}{x} \,:=\, \int_0^1\frac{\sin{tx}}{t}\,dt.$

$\displaystyle {\mathrm{Si}}{z} = z\!-\!\frac{z^3}{3\!\cdot\!3!}\!+\!\frac{z^5}{5\!\cdot\!5!} -\!\frac{z^7}{7\!\cdot\!7!}\!+-\ldots,$

which converges for all complex values

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

$\displaystyle xf'''(x)\!+\!2f''(x)\!+\!xf'(x) = 0.$

$\includegraphics[scale=0.4]{sinint}$

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

Remark 2. There is also another ``sine integral''

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

and the corresponding cosine integral

$\displaystyle {\mathrm{ci}}{x} := \int_\infty^x\frac{\cos t}{t}\,dt = \gamma\!+\ln{x}+\!\int_0^x\frac{\cos{t}\!-\!1}{t}\,dt$

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

"sine integral" is owned by pahio. [ full author list (2) ]

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Other names:

sinus integralis, Si

Also defines:

sine integral, sinus integralis, cosine integral

This object's parent.

Attachments:: sine integral at infinity (Derivation) by pahio
relative of cosine integral (Example) by pahio

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Cross-references: differential equation, entire transcendental function, complex, converges, Taylor series, implies, equation, elementary function, function
There are 11 references to this entry.

This is version 13 of sine integral, born on 2005-03-04, modified 2008-10-04.
Object id is 6844, canonical name is SineIntegral.
Accessed 13140 times total.

Classification:

30A99 (Functions of a complex variable :: General properties :: Miscellaneous)

Pending Errata and Addenda

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