Difference between revisions of "Chi"
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| − | The hyperbolic cosine integral $\mathrm{ | + | The hyperbolic cosine integral $\mathrm{Chi} \colon (0,\infty) \rightarrow \mathbb{R}$ is defined by the formula |
| − | $$\mathrm{ | + | $$\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,$$ |
where $\gamma$ denotes the [[Euler-Mascheroni constant]], $\log$ denotes the [[logarithm]], and $\mathrm{cosh}$ denotes the [[cosh|hyperbolic cosine]] function. | where $\gamma$ denotes the [[Euler-Mascheroni constant]], $\log$ denotes the [[logarithm]], and $\mathrm{cosh}$ denotes the [[cosh|hyperbolic cosine]] function. | ||
<div align="center"> | <div align="center"> | ||
<gallery> | <gallery> | ||
| − | File: | + | File:Chiplot.png|Graph of $\mathrm{chi}$ on $(0,5]$. |
File:Domain coloring hyperbolic cosine integral.png|[[Domain coloring]] of [[analytic continuation]] of $\mathrm{chi}$. | File:Domain coloring hyperbolic cosine integral.png|[[Domain coloring]] of [[analytic continuation]] of $\mathrm{chi}$. | ||
</gallery> | </gallery> | ||
Revision as of 21:55, 23 May 2016
The hyperbolic cosine integral $\mathrm{Chi} \colon (0,\infty) \rightarrow \mathbb{R}$ is defined by the formula $$\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,$$ where $\gamma$ denotes the Euler-Mascheroni constant, $\log$ denotes the logarithm, and $\mathrm{cosh}$ denotes the hyperbolic cosine function.
Graph of $\mathrm{chi}$ on $(0,5]$.
Domain coloring of analytic continuation of $\mathrm{chi}$.
