View source for Integral hyperbolic sine

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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|integral sine]]. The integral hyperbolic sine can be represented by the series

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

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

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

where $\operatorname{Li}$ is the [[Integral logarithm|integral logarithm]].

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

For references see [[Integral cosine|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 $z\in\mathbf C$.

[[Category:Special functions]]