# hyperbolic sine integral

The function (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

 $xf^{\prime\prime\prime}(x)\!+\!2f^{\prime\prime}(x)\!-\!xf^{\prime}(x)=0.$
Title hyperbolic sine integral HyperbolicSineIntegral 2013-03-22 18:27:48 2013-03-22 18:27:48 pahio (2872) pahio (2872) 7 pahio (2872) Definition msc 30A99 Shi HyperbolicFunctions SineIntegral