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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 $,  
+
Functions inverse to the [[hyperbolic functions]]. The inverse hyperbolic functions are the inverse hyperbolic sine, cosine and tangent:  $  \sinh  ^ {-1}  x $,  
$  \cosh  ^ {-} 1 x $,  
+
$  \cosh  ^ {-1}  x $,  
$  \mathop{\rm tanh}  ^ {-} 1 x $;  
+
$  \mathop{\rm tanh}  ^ {-1}  x $;  
 
other notations are:  $  { \mathop{\rm arg}  \sinh }  x $,  
 
other notations are:  $  { \mathop{\rm arg}  \sinh }  x $,  
 
$  { \mathop{\rm arg}  \cosh }  x $,  
 
$  { \mathop{\rm arg}  \cosh }  x $,  
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$$  
 
$$  
\sinh  ^ {-} 1 x  = \  
+
\sinh  ^ {-1}  x  = \  
 
  \mathop{\rm ln} ( x + \sqrt {x  ^ {2} + 1 } ) ,\ \  
 
  \mathop{\rm ln} ( x + \sqrt {x  ^ {2} + 1 } ) ,\ \  
 
- \infty < x < + \infty ,
 
- \infty < x < + \infty ,
Line 28: Line 28:
  
 
$$  
 
$$  
\cosh  ^ {-} 1 x  = \  
+
\cosh  ^ {-1}  x  = \  
 
\pm  \mathop{\rm ln} ( x + \sqrt {x  ^ {2} - 1 } ) ,\ \  
 
\pm  \mathop{\rm ln} ( x + \sqrt {x  ^ {2} - 1 } ) ,\ \  
 
x \geq  1 ,
 
x \geq  1 ,
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$$  
 
$$  
  \mathop{\rm tanh}  ^ {-} 1 x  =   
+
  \mathop{\rm tanh}  ^ {-1}  x  =   
 
\frac{1}{2}
 
\frac{1}{2}
 
   \mathop{\rm ln}   
 
   \mathop{\rm ln}   
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$$
 
$$
  
The inverse hyperbolic functions are single-valued and continuous at each point of their domain of definition, except for  $  \cosh  ^ {-} 1 x $,  
+
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 $
+
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.
 
is chosen, that is, in the formula above only one sign is taken (usually plus). For the graphs of these functions see the figure.
  
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$$  
 
$$  
\sinh  ^ {-} 1 x  = \  
+
\sinh  ^ {-1}  x  = \  
  \mathop{\rm tanh}  ^ {-} 1 \  
+
  \mathop{\rm tanh}  ^ {-1} \  
  
 
\frac{x}{\sqrt {x  ^ {2} + 1 } }
 
\frac{x}{\sqrt {x  ^ {2} + 1 } }
 
  ,\ \  
 
  ,\ \  
  \mathop{\rm tanh}  ^ {-} 1 x  = \  
+
  \mathop{\rm tanh}  ^ {-1}  x  = \  
\sinh  ^ {-} 1 \  
+
\sinh  ^ {-1} \  
  
 
\frac{x}{\sqrt {1 - x  ^ {2} } }
 
\frac{x}{\sqrt {1 - x  ^ {2} } }
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$$  
 
$$  
( \sinh  ^ {-} 1 x )  ^  \prime  = \  
+
( \sinh  ^ {-1}  x )  ^  \prime  = \  
  
 
\frac{1}{\sqrt {x  ^ {2} + 1 } }
 
\frac{1}{\sqrt {x  ^ {2} + 1 } }
 
  ,\ \  
 
  ,\ \  
( \cosh  ^ {-} 1 x )  ^  \prime  =  \pm  
+
( \cosh  ^ {-1}  x )  ^  \prime  =  \pm  
  
 
\frac{1}{\sqrt {x  ^ {2} - 1 } }
 
\frac{1}{\sqrt {x  ^ {2} - 1 } }
Line 79: Line 79:
  
 
$$  
 
$$  
(  \mathop{\rm tanh}  ^ {-} 1 x )  ^  \prime  =   
+
(  \mathop{\rm tanh}  ^ {-1}  x )  ^  \prime  =   
 
\frac{1}{ {1 - x  ^ {2} } }
 
\frac{1}{ {1 - x  ^ {2} } }
 
  .
 
  .
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$$  
 
$$  
\sinh  ^ {-} 1 z  =  - i  { \mathop{\rm arc}  \sin }  i z ,
+
\sinh  ^ {-1}  z  =  - i  { \mathop{\rm arc}  \sin }  i z ,
 
$$
 
$$
  
 
$$  
 
$$  
\cosh  ^ {-} 1 z  =  i  { \mathop{\rm arc}  \cos }  z ,
+
\cosh  ^ {-1}  z  =  i  { \mathop{\rm arc}  \cos }  z ,
 
$$
 
$$
  
 
$$  
 
$$  
  \mathop{\rm tanh}  ^ {-} 1 z  =  - i  { \mathop{\rm arc}  \mathop{\rm tan} }  i z .
+
  \mathop{\rm tanh}  ^ {-1}  z  =  - i  { \mathop{\rm arc}  \mathop{\rm tan} }  i z .
 
$$
 
$$
  

Latest revision as of 19:15, 17 January 2024


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=55169
This article was adapted from an original article by Yu.V. Sudorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article