Difference between revisions of "Integral hyperbolic cosine"
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The special function defined, for real $x$, by | The special function defined, for real $x$, by | ||
| − | $$\operatorname{Chi}(x)=c+\ln x+\int\limits_0^x\frac{\cosh t-1}{t}dt=\operatorname{Ci}(ix)+i\frac\pi2,$$ | + | $$\operatorname{Chi}(x)=c+\ln x+\int\limits_0^x\frac{\cosh t-1}{t}\,dt=\operatorname{Ci}(ix)+i\frac\pi2,$$ |
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 | 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 | ||
| − | $$\operatorname{Chi}(x)=c+\ln+\frac{x^2}{2!2}+\frac{x^4}{4!4}+\ | + | $$\operatorname{Chi}(x)=c+\ln+\frac{x^2}{2!2}+\frac{x^4}{4!4}+\dotsb.$$ |
Sometimes it is denoted by $\chi(x)$. | Sometimes it is denoted by $\chi(x)$. | ||
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One has $\operatorname{Chi}(x)+\operatorname{Shi}(x)=\operatorname{Li}(e^x)$, where $\operatorname{Shi}$ is the [[Integral hyperbolic sine|integral hyperbolic sine]] and $\operatorname{Li}$ is the [[Integral logarithm|integral logarithm]]. | One has $\operatorname{Chi}(x)+\operatorname{Shi}(x)=\operatorname{Li}(e^x)$, where $\operatorname{Shi}$ is the [[Integral hyperbolic sine|integral hyperbolic sine]] and $\operatorname{Li}$ is the [[Integral logarithm|integral logarithm]]. | ||
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| + | [[Category:Special functions]] | ||
Latest revision as of 14:30, 14 February 2020
The special function defined, for real $x$, by
$$\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.
Integral hyperbolic cosine. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integral_hyperbolic_cosine&oldid=33005