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Difference between revisions of "Integral hyperbolic cosine"

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The special function defined, for real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051480/i0514801.png" />, by
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The special function defined, for real $x$, by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051480/i0514802.png" /></td> </tr></table>
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$$\operatorname{Chi}(x)=c+\ln x+\int\limits_0^x\frac{\cosh t-1}{t}dt=\operatorname{Ci}(ix)+i\frac\pi2,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051480/i0514803.png" /> is the [[Euler constant|Euler constant]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051480/i0514804.png" /> is the [[Integral cosine|integral cosine]]. The integral hyperbolic cosine can be represented by the series
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051480/i0514805.png" /></td> </tr></table>
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$$\operatorname{Chi}(x)=c+\ln+\frac{x^2}{2!2}+\frac{x^4}{4!4}+\ldots.$$
  
Sometimes it is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051480/i0514806.png" />.
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Sometimes it is denoted by $\chi(x)$.
  
 
For references, see [[Integral cosine|Integral cosine]].
 
For references, see [[Integral cosine|Integral cosine]].
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====Comments====
 
====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051480/i0514807.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051480/i0514808.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051480/i0514809.png" /> is the [[Integral hyperbolic sine|integral hyperbolic sine]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051480/i05148010.png" /> is the [[Integral logarithm|integral logarithm]].
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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]].

Revision as of 11:12, 19 August 2014

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

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

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.

How to Cite This Entry:
Integral hyperbolic cosine. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integral_hyperbolic_cosine&oldid=12550
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article