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''inverse circular functions''
 
''inverse circular functions''
  
Functions inverse to the [[Trigonometric functions|trigonometric functions]]. The six basic trigonometric functions correspond to the six inverse trigonometric functions. These are called arcussine, arcuscosine, arcustangent, arcuscotangent, arcussecant, arcuscosecant, and are denoted, respectively, by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i0524101.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i0524102.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i0524103.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i0524104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i0524105.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i0524106.png" />. The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i0524107.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i0524108.png" /> are defined (in the real domain) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i0524109.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241011.png" /> for all real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241012.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241014.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241015.png" />; the last two functions are seldom used. Other notations are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241017.png" />, etc.
+
Functions inverse to the [[Trigonometric functions|trigonometric functions]]. The six basic trigonometric functions correspond to the six inverse trigonometric functions. These are called arcussine, arcuscosine, arcustangent, arcuscotangent, arcussecant, arcuscosecant, and are denoted, respectively, by $  { \mathop{\rm Arc}  \sin }  x $,  
 +
$  { \mathop{\rm Arc}  \cos }  x $,  
 +
$  { \mathop{\rm Arc}  \mathop{\rm tan} }  x $,  
 +
$  { \mathop{\rm Arc}  \mathop{\rm cotan} }  x $,  
 +
$  { \mathop{\rm Arc}  \mathop{\rm sec} }  x $,  
 +
$  { \mathop{\rm Arc}  \cosec }  x $.  
 +
The functions $  { \mathop{\rm Arc}  \sin }  x $
 +
and $  { \mathop{\rm Arc}  \cos }  x $
 +
are defined (in the real domain) for $  | x | \leq  1 $;  
 +
$  { \mathop{\rm Arc}  \mathop{\rm tan} }  x $
 +
and $  { \mathop{\rm Arc}  \mathop{\rm cotan} }  x $
 +
for all real $  x $;  
 +
$  { \mathop{\rm Arc}  \mathop{\rm sec} }  x $
 +
and $  { \mathop{\rm Arc}  \cosec }  x $
 +
for $  | x | \geq  1 $;  
 +
the last two functions are seldom used. Other notations are $  \sin  ^ {-} 1  x $,  
 +
$  \cos  ^ {-} 1  x $,  
 +
etc.
  
Since the trigonometric functions are periodic, their inverses are many-valued. The single-valued branches (principal branches) of these functions are denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241019.png" />. Namely, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241020.png" /> is the branch of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241021.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241022.png" />. Similarly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241025.png" /> are defined by the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241028.png" />.
+
Since the trigonometric functions are periodic, their inverses are many-valued. The single-valued branches (principal branches) of these functions are denoted by $  { \mathop{\rm arc}  \sin }  x $,
 +
$  { \mathop{\rm arc}  \cos }  x ,\dots $.  
 +
Namely, $  { \mathop{\rm arc}  \sin }  x $
 +
is the branch of $  { \mathop{\rm Arc}  \sin }  x $
 +
for which $  - \pi / 2 \leq  { \mathop{\rm arc}  \sin }  x \leq  \pi / 2 $.  
 +
Similarly, $  { \mathop{\rm arc}  \cos }  x $,  
 +
$  { \mathop{\rm arc}  \mathop{\rm tan} }  x $
 +
and $  { \mathop{\rm arc}  \mathop{\rm cotan} }  x $
 +
are defined by the conditions 0 \leq  { \mathop{\rm arc}  \cos }  x \leq  \pi $,
 +
$  - \pi / 2 \leq  { \mathop{\rm arc}  \mathop{\rm tan} }  x \leq  \pi / 2 $,  
 +
$  0 < { \mathop{\rm arc}  \mathop{\rm cotan} }  x < \pi $.
  
The figures show the graphs of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241032.png" />; the principal branches are distinguished by a heavy line.
+
The figures show the graphs of $  y = { \mathop{\rm Arc}  \sin }  x $,  
 +
$  y = { \mathop{\rm Arc}  \cos }  x $,  
 +
$  y = { \mathop{\rm Arc}  \mathop{\rm tan} }  x $,  
 +
$  y = { \mathop{\rm Arc}  \mathop{\rm cotan} }  x $;  
 +
the principal branches are distinguished by a heavy line.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/i052410a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/i052410a.gif" />
Line 23: Line 66:
 
Figure: i052410d
 
Figure: i052410d
  
The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241033.png" /> are easily expressed in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241034.png" /> for example:
+
The functions $  { \mathop{\rm Arc}  \sin }  x \dots $
 +
are easily expressed in terms of $  { \mathop{\rm arc}  \sin }  x \dots $
 +
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/i052410/i05241035.png" /></td> </tr></table>
+
$$
 +
{ \mathop{\rm Arc}  \sin }  x  = \
 +
( - 1 )  ^ {n}  { \mathop{\rm arc}  \sin }  x + \pi n ,
 +
$$
  
<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/i052410/i05241036.png" /></td> </tr></table>
+
$$
 +
{ \mathop{\rm Arc}  \cos }  x  = \
 +
\pm  { \mathop{\rm arc}  \cos }  x + 2 \pi n ,
 +
$$
  
<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/i052410/i05241037.png" /></td> </tr></table>
+
$$
 +
{ \mathop{\rm Arc}  \mathop{\rm tan} }  x  = { \mathop{\rm arc}  \mathop{\rm tan} }  x + \pi n ,
 +
$$
  
<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/i052410/i05241038.png" /></td> </tr></table>
+
$$
 +
{ \mathop{\rm Arc}  \mathop{\rm cotan} }  x  = { \mathop{\rm arc}  \mathop{\rm cotan} }  x + \pi n ,
 +
$$
  
<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/i052410/i05241039.png" /></td> </tr></table>
+
$$
 +
= 0 , \pm  1 ,\dots .
 +
$$
  
 
The inverse trigonometric functions are related by
 
The inverse trigonometric functions are related 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/i052/i052410/i05241040.png" /></td> </tr></table>
+
$$
 +
{ \mathop{\rm arc}  \sin }  x + { \mathop{\rm arc}  \cos }  x  = \
 +
 
 +
\frac \pi {2}
 +
,\ \
 +
- 1 \leq  x \leq  1 ,
 +
$$
  
<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/i052410/i05241041.png" /></td> </tr></table>
+
$$
 +
{ \mathop{\rm arc}  \mathop{\rm tan} }  x + { \mathop{\rm arc}  \mathop{\rm cotan} }  x  =
 +
\frac \pi {2}
 +
,\  - \infty < x < + \infty .
 +
$$
  
Hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241043.png" /> are not included in the following formulas.
+
Hence $  { \mathop{\rm arc}  \cos }  x $
 +
and $  { \mathop{\rm arc}  \mathop{\rm cotan} }  x $
 +
are not included in the following formulas.
  
 
The inverse trigonometric functions are infinitely differentiable and can be expanded in a series in a neighbourhood of any interior point of their domain of definition. The derivatives, integrals and series expansions are:
 
The inverse trigonometric functions are infinitely differentiable and can be expanded in a series in a neighbourhood of any interior point of their domain of definition. The derivatives, integrals and series expansions are:
  
<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/i052410/i05241044.png" /></td> </tr></table>
+
$$
 +
( { \mathop{\rm arc}  \sin }  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/i052410/i05241045.png" /></td> </tr></table>
+
\frac{1}{\sqrt {1 - x  ^ {2} } }
 +
,\ \
 +
( { \mathop{\rm arc}  \mathop{\rm tan} }  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/i052410/i05241046.png" /></td> </tr></table>
+
\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/i052410/i05241047.png" /></td> </tr></table>
+
$$
 +
\int\limits { \mathop{\rm arc}  \sin }  x  d x  = x
 +
{ \mathop{\rm arc}  \sin }  x + \sqrt {1 - x  ^ {2} } + C ,
 +
$$
  
<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/i052410/i05241048.png" /></td> </tr></table>
+
$$
 +
\int\limits { \mathop{\rm arc}  \mathop{\rm tan} }  x  d x  = x  { \mathop{\rm arc}  \mathop{\rm tan} }
 +
x -
 +
\frac{1}{2}
 +
  \mathop{\rm ln} ( 1 + x  ^ {2} ) + C ,
 +
$$
 +
 
 +
$$
 +
{ \mathop{\rm arc}  \sin }  x  = x + \sum _ { n= } 1 ^  \infty 
 +
\frac{
 +
( 2 n - 1 ) !! }{( 2 n ) !! }
 +
 +
\frac{x  ^ {2n+} 1 }{2 n + 1 }
 +
,\  | x | < 1 ,
 +
$$
 +
 
 +
$$
 +
{ \mathop{\rm arc}  \mathop{\rm tan} }  x  = \sum _ { n= } 0 ^  \infty 
 +
\frac{( -
 +
1 )  ^ {n} }{2 n + 1 }
 +
x  ^ {2n+} 1 ,\  | x | < 1 .
 +
$$
  
 
The inverse trigonometric functions of a complex variable are defined as the analytic continuations to the complex plane of the corresponding real functions.
 
The inverse trigonometric functions of a complex variable are defined as the analytic continuations to the complex plane of the corresponding real functions.
Line 59: Line 157:
 
The inverse trigonometric functions can be expressed in terms of the [[Logarithmic function|logarithmic function]]:
 
The inverse trigonometric functions can be expressed in terms of the [[Logarithmic function|logarithmic function]]:
  
<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/i052410/i05241049.png" /></td> </tr></table>
+
$$
 
+
{ \mathop{\rm arc}  \sin }  z  = -
<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/i052410/i05241050.png" /></td> </tr></table>
+
i  \mathop{\rm ln} ( i z + \sqrt {1 - z  ^ {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/i052410/i05241051.png" /></td> </tr></table>
 
  
<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/i052410/i05241052.png" /></td> </tr></table>
+
$$
 +
{ \mathop{\rm arc}  \cos }  z  = - i   \mathop{\rm ln} ( z + \sqrt {z  ^ {2} - 1 } ) ,
 +
$$
  
 +
$$
 +
{ \mathop{\rm arc}  \mathop{\rm tan} }  z  =  -
 +
\frac{i}{2}
 +
  \mathop{\rm ln} 
 +
\frac{1 + i z }{1 - i z }
 +
,
 +
$$
  
 +
$$
 +
{ \mathop{\rm arc}  \mathop{\rm cotan} }  z  =  -
 +
\frac{i}{2}
 +
  \mathop{\rm ln} 
 +
\frac{i z - 1 }{i z + 1 }
 +
.
 +
$$
  
 
====Comments====
 
====Comments====
Other notations for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241054.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241056.png" />, respectively.
+
Other notations for $  \mathop{\rm tan}  ^ {-} 1  x $
 +
and $  \mathop{\rm cotan}  ^ {-} 1  x $
 +
are $  \mathop{\rm tg}  ^ {-} 1  x $
 +
and $  \mathop{\rm ctg}  ^ {-} 1  x $,  
 +
respectively.
  
 
====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


inverse circular functions

Functions inverse to the trigonometric functions. The six basic trigonometric functions correspond to the six inverse trigonometric functions. These are called arcussine, arcuscosine, arcustangent, arcuscotangent, arcussecant, arcuscosecant, and are denoted, respectively, by $ { \mathop{\rm Arc} \sin } x $, $ { \mathop{\rm Arc} \cos } x $, $ { \mathop{\rm Arc} \mathop{\rm tan} } x $, $ { \mathop{\rm Arc} \mathop{\rm cotan} } x $, $ { \mathop{\rm Arc} \mathop{\rm sec} } x $, $ { \mathop{\rm Arc} \cosec } x $. The functions $ { \mathop{\rm Arc} \sin } x $ and $ { \mathop{\rm Arc} \cos } x $ are defined (in the real domain) for $ | x | \leq 1 $; $ { \mathop{\rm Arc} \mathop{\rm tan} } x $ and $ { \mathop{\rm Arc} \mathop{\rm cotan} } x $ for all real $ x $; $ { \mathop{\rm Arc} \mathop{\rm sec} } x $ and $ { \mathop{\rm Arc} \cosec } x $ for $ | x | \geq 1 $; the last two functions are seldom used. Other notations are $ \sin ^ {-} 1 x $, $ \cos ^ {-} 1 x $, etc.

Since the trigonometric functions are periodic, their inverses are many-valued. The single-valued branches (principal branches) of these functions are denoted by $ { \mathop{\rm arc} \sin } x $, $ { \mathop{\rm arc} \cos } x ,\dots $. Namely, $ { \mathop{\rm arc} \sin } x $ is the branch of $ { \mathop{\rm Arc} \sin } x $ for which $ - \pi / 2 \leq { \mathop{\rm arc} \sin } x \leq \pi / 2 $. Similarly, $ { \mathop{\rm arc} \cos } x $, $ { \mathop{\rm arc} \mathop{\rm tan} } x $ and $ { \mathop{\rm arc} \mathop{\rm cotan} } x $ are defined by the conditions $ 0 \leq { \mathop{\rm arc} \cos } x \leq \pi $, $ - \pi / 2 \leq { \mathop{\rm arc} \mathop{\rm tan} } x \leq \pi / 2 $, $ 0 < { \mathop{\rm arc} \mathop{\rm cotan} } x < \pi $.

The figures show the graphs of $ y = { \mathop{\rm Arc} \sin } x $, $ y = { \mathop{\rm Arc} \cos } x $, $ y = { \mathop{\rm Arc} \mathop{\rm tan} } x $, $ y = { \mathop{\rm Arc} \mathop{\rm cotan} } x $; the principal branches are distinguished by a heavy line.

Figure: i052410a

Figure: i052410b

Figure: i052410c

Figure: i052410d

The functions $ { \mathop{\rm Arc} \sin } x \dots $ are easily expressed in terms of $ { \mathop{\rm arc} \sin } x \dots $ for example:

$$ { \mathop{\rm Arc} \sin } x = \ ( - 1 ) ^ {n} { \mathop{\rm arc} \sin } x + \pi n , $$

$$ { \mathop{\rm Arc} \cos } x = \ \pm { \mathop{\rm arc} \cos } x + 2 \pi n , $$

$$ { \mathop{\rm Arc} \mathop{\rm tan} } x = { \mathop{\rm arc} \mathop{\rm tan} } x + \pi n , $$

$$ { \mathop{\rm Arc} \mathop{\rm cotan} } x = { \mathop{\rm arc} \mathop{\rm cotan} } x + \pi n , $$

$$ n = 0 , \pm 1 ,\dots . $$

The inverse trigonometric functions are related by

$$ { \mathop{\rm arc} \sin } x + { \mathop{\rm arc} \cos } x = \ \frac \pi {2} ,\ \ - 1 \leq x \leq 1 , $$

$$ { \mathop{\rm arc} \mathop{\rm tan} } x + { \mathop{\rm arc} \mathop{\rm cotan} } x = \frac \pi {2} ,\ - \infty < x < + \infty . $$

Hence $ { \mathop{\rm arc} \cos } x $ and $ { \mathop{\rm arc} \mathop{\rm cotan} } x $ are not included in the following formulas.

The inverse trigonometric functions are infinitely differentiable and can be expanded in a series in a neighbourhood of any interior point of their domain of definition. The derivatives, integrals and series expansions are:

$$ ( { \mathop{\rm arc} \sin } x ) ^ \prime = \ \frac{1}{\sqrt {1 - x ^ {2} } } ,\ \ ( { \mathop{\rm arc} \mathop{\rm tan} } x ) ^ \prime = \ \frac{1}{1 + x ^ {2} } , $$

$$ \int\limits { \mathop{\rm arc} \sin } x d x = x { \mathop{\rm arc} \sin } x + \sqrt {1 - x ^ {2} } + C , $$

$$ \int\limits { \mathop{\rm arc} \mathop{\rm tan} } x d x = x { \mathop{\rm arc} \mathop{\rm tan} } x - \frac{1}{2} \mathop{\rm ln} ( 1 + x ^ {2} ) + C , $$

$$ { \mathop{\rm arc} \sin } x = x + \sum _ { n= } 1 ^ \infty \frac{ ( 2 n - 1 ) !! }{( 2 n ) !! } \frac{x ^ {2n+} 1 }{2 n + 1 } ,\ | x | < 1 , $$

$$ { \mathop{\rm arc} \mathop{\rm tan} } x = \sum _ { n= } 0 ^ \infty \frac{( - 1 ) ^ {n} }{2 n + 1 } x ^ {2n+} 1 ,\ | x | < 1 . $$

The inverse trigonometric functions of a complex variable are defined as the analytic continuations to the complex plane of the corresponding real functions.

The inverse trigonometric functions can be expressed in terms of the logarithmic function:

$$ { \mathop{\rm arc} \sin } z = - i \mathop{\rm ln} ( i z + \sqrt {1 - z ^ {2} } ) , $$

$$ { \mathop{\rm arc} \cos } z = - i \mathop{\rm ln} ( z + \sqrt {z ^ {2} - 1 } ) , $$

$$ { \mathop{\rm arc} \mathop{\rm tan} } z = - \frac{i}{2} \mathop{\rm ln} \frac{1 + i z }{1 - i z } , $$

$$ { \mathop{\rm arc} \mathop{\rm cotan} } z = - \frac{i}{2} \mathop{\rm ln} \frac{i z - 1 }{i z + 1 } . $$

Comments

Other notations for $ \mathop{\rm tan} ^ {-} 1 x $ and $ \mathop{\rm cotan} ^ {-} 1 x $ are $ \mathop{\rm tg} ^ {-} 1 x $ and $ \mathop{\rm ctg} ^ {-} 1 x $, respectively.

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 trigonometric functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Inverse_trigonometric_functions&oldid=47423
This article was adapted from an original article by Yu.V. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article