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'sine integral'
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| Title of object: |
sine integral |
| Canonical Name: |
SineIntegral |
| Type: |
Definition |
| Created on: |
2005-03-04 13:07:35 |
| Modified on: |
2005-03-04 14:05:35 |
| Classification: |
msc:30A99 |
| Synonyms: |
sine integral=sinus integralis |
Preamble:
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Content:
The function {\em sine integral} (in Latin {\em sinus integralis}) is defined as
$$\mathrm{Si}(x) := \int_0^x\frac{\sin t}{t}dt.$$
So it has the Taylor series \PMlinkescapetext{expansion}
$$\mathrm{Si}(z) = z-\frac{z^3}{3\cdot 3!}+\frac{z^5}{5\cdot 5!}
-\frac{z^7}{7\cdot 7!}+-...$$
which converges for all complex values $z$, and thus defines an entire transcendental function.
$\mathrm{Si}(x)$ satisfies the linear third \PMlinkescapetext{order} differential equation
$$xf'''(x)+2f''(x)+xf'(x) = 0.$$ |
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