| Version 3 |
Version 2 |
| The function {\em hyperbolic sine integral} (in Latin {\em sinus hyperbolicus integralis}) from $\mathbb{R}$ to $\mathbb{R}$ is defined as |
The function {\em hyperbolic sine integral} (in Latin {\em sinus hyperbolicus integralis}) from $\mathbb{R}$ to $\mathbb{R}$ is defined as |
| \begin{align} |
$$\Shi{x} \,:=\, \int_0^x\frac{\sinh t}{t}\,dt,$$ |
| \Shi{x} \,:=\, \int_0^x\frac{\sinh t}{t}\,dt, |
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| \end{align} |
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| or alternatively as |
or alternatively as |
| $$\Shi{x} \,:=\, \int_0^1\frac{\sinh{tx}}{t}\,dt.$$\\ |
$$\Shi{x} \,:=\, \int_0^1\frac{\sinh{tx}}{t}\,dt.$$\\ |
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It isn't an elementary function.\, The equation (1) implies the Taylor series \PMlinkescapetext{expansion}
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So the function has the Taylor series \PMlinkescapetext{expansion}
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| $$\Shi{z} = z\!+\!\frac{z^3}{3\!\cdot\!3!}\!+\!\frac{z^5}{5\!\cdot\!5!} |
$$\Shi{z} = z\!+\!\frac{z^3}{3\!\cdot\!3!}\!+\!\frac{z^5}{5\!\cdot\!5!} |
| \!+\!\frac{z^7}{7\!\cdot\!7!}\!+\cdots,$$ |
\!+\!\frac{z^7}{7\!\cdot\!7!}\!+\cdots,$$ |
| which converges for all complex values $z$ and thus defines an entire transcendental function.\\ |
which converges for all complex values $z$ and thus defines an entire transcendental function.\\ |
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| $\Shi{x}$ satisfies the linear third \PMlinkescapetext{order} differential equation |
$\Shi{x}$ satisfies the linear third \PMlinkescapetext{order} differential equation |
| $$xf'''(x)\!+\!2f''(x)\!-\!xf'(x) = 0.$$ |
$$xf'''(x)\!+\!2f''(x)\!-\!xf'(x) = 0.$$ |
