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Functions inverse to the [[Hyperbolic functions|hyperbolic functions]]. The inverse hyperbolic functions are the inverse hyperbolic sine, cosine and tangent: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052370/i0523701.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052370/i0523702.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052370/i0523703.png" />; other notations are: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052370/i0523704.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052370/i0523705.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052370/i0523706.png" />.
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The inverse hyperbolic functions of a real variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052370/i0523707.png" /> are defined by the formulas
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{{TEX|auto}}
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{{TEX|done}}
  
<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/i052/i052370/i0523708.png" /></td> </tr></table>
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Functions inverse to the [[Hyperbolic functions|hyperbolic functions]]. The inverse hyperbolic functions are the inverse hyperbolic sine, cosine and tangent: $  \sinh  ^ {-} 1  x $,
 +
$  \cosh  ^ {-} 1  x $,
 +
$  \mathop{\rm tanh}  ^ {-} 1  x $;  
 +
other notations are: $  { \mathop{\rm arg}  \sinh }  x $,
 +
$  { \mathop{\rm arg}  \cosh }  x $,
 +
$  { \mathop{\rm arg}  \mathop{\rm tanh} }  x $.
  
<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/i052/i052370/i0523709.png" /></td> </tr></table>
+
The inverse hyperbolic functions of a real variable  $  x $
 +
are defined by the formulas
  
<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/i052/i052370/i05237010.png" /></td> </tr></table>
+
$$
 +
\sinh  ^ {-} 1  x  = \
 +
\mathop{\rm ln} ( x + \sqrt {x  ^ {2} + 1 } ) ,\ \
 +
- \infty < x < + \infty ,
 +
$$
  
The inverse hyperbolic functions are single-valued and continuous at each point of their domain of definition, except for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052370/i05237011.png" />, which is two-valued. In studying the properties of the inverse hyperbolic functions, one of the continuous branches of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052370/i05237012.png" /> is chosen, that is, in the formula above only one sign is taken (usually plus). For the graphs of these functions see the figure.
+
$$
 +
\cosh  ^ {-} 1  x  = \
 +
\pm  \mathop{\rm ln} ( x + \sqrt {x  ^ {2} - 1 } ) ,\ \
 +
x \geq  1 ,
 +
$$
 +
 
 +
$$
 +
\mathop{\rm tanh}  ^ {-} 1  x  = 
 +
\frac{1}{2}
 +
  \mathop{\rm ln} 
 +
\frac{1
 +
+ x }{1 - x }
 +
,\  | x | < 1 .
 +
$$
 +
 
 +
The inverse hyperbolic functions are single-valued and continuous at each point of their domain of definition, except for $  \cosh  ^ {-} 1  x $,  
 +
which is two-valued. In studying the properties of the inverse hyperbolic functions, one of the continuous branches of $  \cosh  ^ {-} 1  x $
 +
is chosen, that is, in the formula above only one sign is taken (usually plus). For the graphs of these functions see the figure.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/i052370a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/i052370a.gif" />
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There a number of relations between the inverse hyperbolic functions. For example,
 
There a number of relations between the inverse hyperbolic functions. For example,
  
<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/i052/i052370/i05237013.png" /></td> </tr></table>
+
$$
 +
\sinh  ^ {-} 1  x  = \
 +
\mathop{\rm tanh}  ^ {-} 1 \
 +
 
 +
\frac{x}{\sqrt {x  ^ {2} + 1 } }
 +
,\ \
 +
\mathop{\rm tanh}  ^ {-} 1  x  = \
 +
\sinh  ^ {-} 1 \
 +
 
 +
\frac{x}{\sqrt {1 - x  ^ {2} } }
 +
.
 +
$$
  
 
The derivatives of the inverse hyperbolic functions are given by the formulas
 
The derivatives of the inverse hyperbolic functions are given by the formulas
  
<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/i052/i052370/i05237014.png" /></td> </tr></table>
+
$$
 +
( \sinh  ^ {-} 1  x )  ^  \prime  = \
  
<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/i052/i052370/i05237015.png" /></td> </tr></table>
+
\frac{1}{\sqrt {x  ^ {2} + 1 } }
 +
,\ \
 +
( \cosh  ^ {-} 1  x )  ^  \prime  = \pm
  
The inverse hyperbolic functions of a complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052370/i05237016.png" /> are defined by the same formulas as those for a real variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052370/i05237017.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052370/i05237018.png" /> is understood to be the many-valued logarithmic function. The inverse hyperbolic functions of a complex variable are the analytic continuations to the complex plane of the corresponding functions of a real variable.
+
\frac{1}{\sqrt {x  ^ {2} - 1 } }
 +
,
 +
$$
  
The inverse hyperbolic functions can be expressed in terms of the [[Inverse trigonometric functions|inverse trigonometric functions]] by the formulas
+
$$
 +
(  \mathop{\rm tanh}  ^ {-} 1  x )  ^  \prime  = 
 +
\frac{1}{ {1 - x  ^ {2} } }
 +
.
 +
$$
  
<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/i052/i052370/i05237019.png" /></td> </tr></table>
+
The inverse hyperbolic functions of a complex variable  $  z $
 +
are defined by the same formulas as those for a real variable  $  x $,
 +
where  $  \mathop{\rm ln}  z $
 +
is understood to be the many-valued logarithmic function. The inverse hyperbolic functions of a complex variable are the analytic continuations to the complex plane of the corresponding functions of a real variable.
  
<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/i052/i052370/i05237020.png" /></td> </tr></table>
+
The inverse hyperbolic functions can be expressed in terms of the [[Inverse trigonometric functions|inverse trigonometric functions]] by the formulas
  
<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/i052/i052370/i05237021.png" /></td> </tr></table>
+
$$
 +
\sinh  ^ {-} 1  z  = - i { \mathop{\rm arc}  \sin }  i z ,
 +
$$
  
 +
$$
 +
\cosh  ^ {-} 1  z  =  i  { \mathop{\rm arc}  \cos }  z ,
 +
$$
  
 +
$$
 +
\mathop{\rm tanh}  ^ {-} 1  z  =  - i  { \mathop{\rm arc}  \mathop{\rm tan} }  i z .
 +
$$
  
 
====Comments====
 
====Comments====
The notations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052370/i05237022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052370/i05237023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052370/i05237024.png" /> are also quite common.
+
The notations $  { \mathop{\rm arc}  \sinh }  x $,  
 +
$  { \mathop{\rm arc}  \cosh }  x $
 +
and $  { \mathop{\rm arc}  \mathop{\rm tanh} }  x $
 +
are also quite common.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.R. Spiegel,  "Complex variables" , ''Schaum's Outline Series'' , McGraw-Hill  (1974)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Abramowitz,  I.A. Stegun,  "Handbook of mathematical functions" , Dover, reprint  (1972)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.R. Spiegel,  "Complex variables" , ''Schaum's Outline Series'' , McGraw-Hill  (1974)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Abramowitz,  I.A. Stegun,  "Handbook of mathematical functions" , Dover, reprint  (1972)</TD></TR></table>

Latest revision as of 22:13, 5 June 2020


Functions inverse to the hyperbolic functions. The inverse hyperbolic functions are the inverse hyperbolic sine, cosine and tangent: $ \sinh ^ {-} 1 x $, $ \cosh ^ {-} 1 x $, $ \mathop{\rm tanh} ^ {-} 1 x $; other notations are: $ { \mathop{\rm arg} \sinh } x $, $ { \mathop{\rm arg} \cosh } x $, $ { \mathop{\rm arg} \mathop{\rm tanh} } x $.

The inverse hyperbolic functions of a real variable $ x $ are defined by the formulas

$$ \sinh ^ {-} 1 x = \ \mathop{\rm ln} ( x + \sqrt {x ^ {2} + 1 } ) ,\ \ - \infty < x < + \infty , $$

$$ \cosh ^ {-} 1 x = \ \pm \mathop{\rm ln} ( x + \sqrt {x ^ {2} - 1 } ) ,\ \ x \geq 1 , $$

$$ \mathop{\rm tanh} ^ {-} 1 x = \frac{1}{2} \mathop{\rm ln} \frac{1 + x }{1 - x } ,\ | x | < 1 . $$

The inverse hyperbolic functions are single-valued and continuous at each point of their domain of definition, except for $ \cosh ^ {-} 1 x $, which is two-valued. In studying the properties of the inverse hyperbolic functions, one of the continuous branches of $ \cosh ^ {-} 1 x $ is chosen, that is, in the formula above only one sign is taken (usually plus). For the graphs of these functions see the figure.

Figure: i052370a

There a number of relations between the inverse hyperbolic functions. For example,

$$ \sinh ^ {-} 1 x = \ \mathop{\rm tanh} ^ {-} 1 \ \frac{x}{\sqrt {x ^ {2} + 1 } } ,\ \ \mathop{\rm tanh} ^ {-} 1 x = \ \sinh ^ {-} 1 \ \frac{x}{\sqrt {1 - x ^ {2} } } . $$

The derivatives of the inverse hyperbolic functions are given by the formulas

$$ ( \sinh ^ {-} 1 x ) ^ \prime = \ \frac{1}{\sqrt {x ^ {2} + 1 } } ,\ \ ( \cosh ^ {-} 1 x ) ^ \prime = \pm \frac{1}{\sqrt {x ^ {2} - 1 } } , $$

$$ ( \mathop{\rm tanh} ^ {-} 1 x ) ^ \prime = \frac{1}{ {1 - x ^ {2} } } . $$

The inverse hyperbolic functions of a complex variable $ z $ are defined by the same formulas as those for a real variable $ x $, where $ \mathop{\rm ln} z $ is understood to be the many-valued logarithmic function. The inverse hyperbolic functions of a complex variable are the analytic continuations to the complex plane of the corresponding functions of a real variable.

The inverse hyperbolic functions can be expressed in terms of the inverse trigonometric functions by the formulas

$$ \sinh ^ {-} 1 z = - i { \mathop{\rm arc} \sin } i z , $$

$$ \cosh ^ {-} 1 z = i { \mathop{\rm arc} \cos } z , $$

$$ \mathop{\rm tanh} ^ {-} 1 z = - i { \mathop{\rm arc} \mathop{\rm tan} } i z . $$

Comments

The notations $ { \mathop{\rm arc} \sinh } x $, $ { \mathop{\rm arc} \cosh } x $ and $ { \mathop{\rm arc} \mathop{\rm tanh} } x $ are also quite common.

References

[a1] M.R. Spiegel, "Complex variables" , Schaum's Outline Series , McGraw-Hill (1974)
[a2] M. Abramowitz, I.A. Stegun, "Handbook of mathematical functions" , Dover, reprint (1972)
How to Cite This Entry:
Inverse hyperbolic functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Inverse_hyperbolic_functions&oldid=12740
This article was adapted from an original article by Yu.V. Sudorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article