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 | The hyperbolic cosine integral $\mathrm{chi} \colon (0,\infty) \rightarrow \mathbb{R}$ is defined by the formula | ||
| − | $$\mathrm{chi}(z)=\gamma + \log(z) + \displaystyle\int_0^z \dfrac{\mathrm{cosh}(t)-1}{t} | + | $$\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,$$ |
where $\gamma$ denotes the [[Euler-Mascheroni constant]], $\log$ denotes the [[logarithm]], and $\mathrm{cosh}$ denotes the [[cosh|hyperbolic cosine]] function. | where $\gamma$ denotes the [[Euler-Mascheroni constant]], $\log$ denotes the [[logarithm]], and $\mathrm{cosh}$ denotes the [[cosh|hyperbolic cosine]] function. | ||
Revision as of 21:49, 23 May 2016
The hyperbolic cosine integral $\mathrm{chi} \colon (0,\infty) \rightarrow \mathbb{R}$ is defined by the formula $$\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.
- Coshintegral.png
Graph of $\mathrm{chi}$ on $(0,5]$.
Domain coloring of analytic continuation of $\mathrm{chi}$.
