View source for Integral hyperbolic cosine

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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|Euler constant]] and $\operatorname{Ci}(x)$ is the [[Integral cosine|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|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|integral hyperbolic sine]] and $\operatorname{Li}$ is the [[Integral logarithm|integral logarithm]].

[[Category:Special functions]]