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Difference between revisions of "Integral hyperbolic sine"

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The special function defined, for real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051490/i0514901.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/i051490/i0514902.png" /></td> </tr></table>
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$$\operatorname{Shi}(x)=\int\limits_0^x\frac{\sinh t}{t}\,dt=i\operatorname{Si}(ix),$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051490/i0514903.png" /> is the [[Integral sine|integral sine]]. The integral hyperbolic sine can be represented by the series
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where $\operatorname{Si}(x)$ is the [[Integral sine|integral sine]]. The integral hyperbolic sine 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/i051490/i0514904.png" /></td> </tr></table>
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$$\operatorname{Shi}(x)=x+\frac{x^3}{3!3}+\frac{x^5}{5!5}+\dotsb.$$
  
It is related to the [[Integral hyperbolic cosine|integral hyperbolic cosine]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051490/i0514905.png" /> by
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It is related to the [[Integral hyperbolic cosine|integral hyperbolic cosine]] $\operatorname{Chi}(x)$ 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/i051490/i0514906.png" /></td> </tr></table>
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$$\operatorname{Chi}(x)+\operatorname{Shi}(x)=\operatorname{Li}(e^x),$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051490/i0514907.png" /> is the [[Integral logarithm|integral logarithm]].
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where $\operatorname{Li}$ is the [[Integral logarithm|integral logarithm]].
  
Sometimes it is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051490/i0514908.png" />.
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Sometimes it is denoted by $\operatorname{shi}(x)$.
  
 
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 sine integral, is also called the hyperbolic sine integral. It can, of course, be defined (as above) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051490/i0514909.png" />.
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This function, which is seldom used because of its relation with the sine integral, is also called the hyperbolic sine integral. It can, of course, be defined (as above) for $z\in\mathbf C$.
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[[Category:Special functions]]

Latest revision as of 20:40, 1 January 2019

The special function defined, for real $x$, by

$$\operatorname{Shi}(x)=\int\limits_0^x\frac{\sinh t}{t}\,dt=i\operatorname{Si}(ix),$$

where $\operatorname{Si}(x)$ is the integral sine. The integral hyperbolic sine can be represented by the series

$$\operatorname{Shi}(x)=x+\frac{x^3}{3!3}+\frac{x^5}{5!5}+\dotsb.$$

It is related to the integral hyperbolic cosine $\operatorname{Chi}(x)$ by

$$\operatorname{Chi}(x)+\operatorname{Shi}(x)=\operatorname{Li}(e^x),$$

where $\operatorname{Li}$ is the integral logarithm.

Sometimes it is denoted by $\operatorname{shi}(x)$.

For references see Integral cosine.


Comments

This function, which is seldom used because of its relation with the sine integral, is also called the hyperbolic sine integral. It can, of course, be defined (as above) for $z\in\mathbf C$.

How to Cite This Entry:
Integral hyperbolic sine. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integral_hyperbolic_sine&oldid=11472
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article