hyperbolic sine integral

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

$\mathrm{Shi}x:={\displaystyle {\int }_0^x}{\displaystyle \frac{\sinh t}{t}}𝑑t,$(1)

or alternatively as

$$\mathrm{Shi}x:={\int }_0^1\frac{\sinh tx}{t}𝑑t.$$

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

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

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

$$\mathrm{Shi}x=i\mathrm{Si}ix$$

connects Shi to the sine integral function.

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

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