Integral hyperbolic sine

The special function defined, for real $x$, by

$$\operatorname{Shi}(x)=\int\limits_0^x\frac{\sinh t}{t}\,dt=i\operatorname{Si}(ix),$$

where $\operatorname{Si}(x)$ is the integral sine. The integral hyperbolic sine can be represented by the series

$$\operatorname{Shi}(x)=x+\frac{x^3}{3!3}+\frac{x^5}{5!5}+\dotsb.$$

It is related to the integral hyperbolic cosine $\operatorname{Chi}(x)$ by

$$\operatorname{Chi}(x)+\operatorname{Shi}(x)=\operatorname{Li}(e^x),$$

where $\operatorname{Li}$ is the integral logarithm.

Sometimes it is denoted by $\operatorname{shi}(x)$.

For references see Integral cosine.

Comments

This function, which is seldom used because of its relation with the sine integral, is also called the hyperbolic sine integral. It can, of course, be defined (as above) for $z\in\mathbf C$.