hyperbolic sine integralHyperbolicSineIntegral2008-10-03 16:33:062008-10-04 14:42:08Definitioncomplex function% this is the default PlanetMath preamble. as your knowledge
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\DeclareMathOperator{\Shi}{Shi}
\DeclareMathOperator{\Si}{Si}
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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,
\end{align}
or alternatively as
$$\Shi{x} \,:=\, \int_0^1\frac{\sinh{tx}}{t}\,dt.$$\\
It isn't an elementary function.\, The equation (1) implies the Taylor series expansion
$$\Shi{z} = z\!+\!\frac{z^3}{3\!\cdot\!3!}\!+\!\frac{z^5}{5\!\cdot\!5!}
\!+\!\frac{z^7}{7\!\cdot\!7!}\!+\cdots,$$
which converges for all complex values $z$ and thus defines an entire transcendental function.\,
Using the Taylor expansions, it is easily seen that
$$\Shi x \;=\; i\,\Si{ix}$$
connects Shi to the sine integral function.\\
$\Shi{x}$ satisfies the linear third \PMlinkescapetext{order} differential equation
$$xf'''(x)\!+\!2f''(x)\!-\!xf'(x) = 0.$$