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sine integral

(Definition)

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

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

or alternatively as

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

So the function has the Taylor series expansion

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

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

Also defines:

sine integral, sinus integralis, cosine integral

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Cross-references: differential equation, entire transcendental function, complex, converges, Taylor series, Laplace transform of sine integral, function
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This is version 11 of sine integral, born on 2005-03-04, modified 2008-05-15.
Object id is 6844, canonical name is SineIntegral.
Accessed 7871 times total.

Classification:

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

Pending Errata and Addenda

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