Integral hyperbolic cosine

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

$$\operatorname{Chi}(x)=c+\ln x+\int\limits_0^x\frac{\cosh t-1}{t}\,dt=\operatorname{Ci}(ix)+i\frac\pi2,$$

where $c=0.5772\ldots$ is the Euler constant and $\operatorname{Ci}(x)$ is the integral cosine. The integral hyperbolic cosine can be represented by the series

$$\operatorname{Chi}(x)=c+\ln+\frac{x^2}{2!2}+\frac{x^4}{4!4}+\dotsb.$$

Sometimes it is denoted by $\chi(x)$.

For references, see Integral cosine.

Comments

This function, which is seldom used because of its relation with the cosine integral, is also called the hyperbolic cosine integral. It can, of course be defined (as above) for $z\in\mathbf C\setminus\{x\in\mathbf R\colon x\leq0\}$.

One has $\operatorname{Chi}(x)+\operatorname{Shi}(x)=\operatorname{Li}(e^x)$, where $\operatorname{Shi}$ is the integral hyperbolic sine and $\operatorname{Li}$ is the integral logarithm.