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 and $\operatorname{Ci}(x)$ is the integral cosine. The integral hyperbolic cosine can be represented by the series


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

For references, see Integral cosine.


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.

How to Cite This Entry:
Integral hyperbolic cosine. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article