hyperbolic sine integral

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

$\displaystyle\operatorname{Shi}{x}\,:=\,\int_{0}^{x}\frac{\sinh t}{t}\,dt,$

(1)

or alternatively as

$\operatorname{Shi}{x}\,:=\,\int_{0}^{1}\frac{\sinh{tx}}{t}\,dt.$

It isn’t an elementary function. The equation (1) implies the Taylor series expansion

$\operatorname{Shi}{z}=z\!+\!\frac{z^{3}}{3\!\cdot\!3!}\!+\!\frac{z^{5}}{5\!% \cdot\!5!}\!+\!\frac{z^{7}}{7\!\cdot\!7!}\!+\cdots,$

which converges for all complex values $z$ and thus defines an entire transcendental function. Using the Taylor expansions, it is easily seen that

$\operatorname{Shi}x\;=\;i\,\operatorname{Si}{ix}$

connects Shi to the sine integral function.

$\operatorname{Shi}{x}$ satisfies the linear third differential equation

$xf^{\prime\prime\prime}(x)\!+\!2f^{\prime\prime}(x)\!-\!xf^{\prime}(x)=0.$