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Version 1 |
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The function {\em sine integral} (in Latin {\em sinus integralis}) is defined as
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The function {\em sine integral} (in Latin {\em sinus integralis}) is usually defined as
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$$\mathrm{Si}(x) := \int_0^x\frac{\sin t}{t}dt.$$
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$$\mathrm{Si}{z} := \int_0^z\frac{\sin t}{t}dt.$$
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| So it has the Taylor series \PMlinkescapetext{expansion} |
So it has the Taylor series \PMlinkescapetext{expansion} |
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$$\mathrm{Si}(z) = z-\frac{z^3}{3\cdot 3!}+\frac{z^5}{5\cdot 5!}
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$$\mathrm{Si}{z} = z-\frac{z^3}{3\cdot 3!}+\frac{z^5}{5\cdot 5!}
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| -\frac{z^7}{7\cdot 7!}+-...$$ |
-\frac{z^7}{7\cdot 7!}+-...$$ |
| which converges for all complex values $z$, and thus defines an entire transcendental function. |
This converges for all complex values $z$, and thus defines an entire transcendental function. |
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| $\mathrm{Si}(x)$ satisfies the linear third \PMlinkescapetext{order} differential equation |
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| $$xf'''(x)+2f''(x)+xf'(x) = 0.$$ |
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