Difference between revisions of "Chi"

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The hyperbolic cosine integral $\mathrm{Chi} \colon (0,\infty) \rightarrow \mathbb{R}$ is defined by the formula
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The hyperbolic cosine integral $\mathrm{Chi}$ is defined for $|\mathrm{arg}(z)| < \pi$ the formula
$$\mathrm{Chi}(z)=-\displaystyle\int_z^{\infty} \dfrac{\mathrm{cosh}(t)}{t} \mathrm{d}t=\gamma + \log(z) + \displaystyle\int_0^z \dfrac{\mathrm{cosh}(t)-1}{t} \mathrm{d}t,$$
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$$\mathrm{Chi}(z)=\gamma + \log(z) + \displaystyle\int_0^z \dfrac{\cosh(t)-1}{t} \mathrm{d}t,$$
where $\gamma$ denotes the [[Euler-Mascheroni constant]], $\log$ denotes the [[logarithm]], and $\mathrm{cosh}$ denotes the [[cosh|hyperbolic cosine]] function.
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where $\gamma$ denotes the [[Euler-Mascheroni constant]], $\log$ denotes the [[logarithm]], and $\cosh$ denotes the [[cosh|hyperbolic cosine]].
  
 
<div align="center">
 
<div align="center">
 
<gallery>
 
<gallery>
File:Chiplot.png|Graph of $\mathrm{Chi}$ on $(0,5]$.
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File:Chiplot.png|Graph of $\mathrm{Chi}$.
 
File:Complexchiplot.png|[[Domain coloring]] of $\mathrm{Chi}$.
 
File:Complexchiplot.png|[[Domain coloring]] of $\mathrm{Chi}$.
 
</gallery>
 
</gallery>
 
</div>
 
</div>
  
<center>{{:*-integral functions footer}}</center>
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=Properties=
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[[Derivative of chi]]<br />
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[[Antiderivative of chi]]<br />
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{{:*-integral functions footer}}
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[[Category:SpecialFunction]]

Latest revision as of 23:57, 10 December 2016

The hyperbolic cosine integral $\mathrm{Chi}$ is defined for $|\mathrm{arg}(z)| < \pi$ the formula $$\mathrm{Chi}(z)=\gamma + \log(z) + \displaystyle\int_0^z \dfrac{\cosh(t)-1}{t} \mathrm{d}t,$$ where $\gamma$ denotes the Euler-Mascheroni constant, $\log$ denotes the logarithm, and $\cosh$ denotes the hyperbolic cosine.

Properties

Derivative of chi
Antiderivative of chi

$\ast$-integral functions