Difference between revisions of "Arccos"

From specialfunctionswiki
Jump to: navigation, search
 
(28 intermediate revisions by the same user not shown)
Line 1: Line 1:
The $\mathrm{arccos}$ function is the inverse function of the [[cosine]] function.
+
__NOTOC__
 +
The function $\mathrm{arccos} \colon \mathbb{C} \setminus \{(-\infty,-1) \bigcup (1,\infty) \} \rightarrow \mathbb{C}$ is defined by
 +
$$\rm{arccos}(z)=\dfrac{\pi}{2} + i\log\left( iz + \sqrt{1-z^2} \right),$$
 +
where $i$ denotes the [[imaginary number]] and $\log$ denotes the [[logarithm]].
  
[[File:Arccos.png|500px]]
+
<div align="center">
 +
<gallery>
 +
File:Arccosplot.png|Graph of $\mathrm{arccos}$ on $[-1,1]$.
 +
File:Complexarccosplot.png|[[Domain coloring]] of $\mathrm{arccos}$.
 +
</gallery>
 +
</div>
  
 
=Properties=
 
=Properties=
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
+
[[Arccos as inverse cosine]]<br />
<strong>Proposition:</strong>  
+
[[Derivative of arccos]]<br />
$$\dfrac{d}{dz} \mathrm{arccos}(z) = -\dfrac{1}{\sqrt{1-z^2}}$$
+
[[Antiderivative of arccos]]<br />
<div class="mw-collapsible-content">
+
 
<strong>Proof:</strong> █
+
=References=
</div>
+
[http://mathworld.wolfram.com/InverseCosine.html  Weisstein, Eric W. "Inverse Cosine." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/InverseCosine.html]
</div>
 
  
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
+
=See Also=
<strong>Proposition:</strong>
+
[[Cosine]] <br />
$$\int \mathrm{arccos}(z) dz = z\mathrm{arccos}(z)-\sqrt{1-z^2}+C$$
+
[[Cosh]] <br />
<div class="mw-collapsible-content">
+
[[Arccosh]]
<strong>Proof:</strong>
 
</div>
 
</div>
 
  
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
+
{{:Inverse trigonometric functions footer}}
<strong>Proposition:</strong>
 
$$\mathrm{arccos}(z)=\mathrm{arcsec} \left( \dfrac{1}{z} \right)$$
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
</div>
 
</div>
 
  
=References=
+
[[Category:SpecialFunction]]
[http://mathworld.wolfram.com/InverseCosine.html  Weisstein, Eric W. "Inverse Cosine." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/InverseCosine.html]
 

Latest revision as of 20:04, 22 November 2016

The function $\mathrm{arccos} \colon \mathbb{C} \setminus \{(-\infty,-1) \bigcup (1,\infty) \} \rightarrow \mathbb{C}$ is defined by $$\rm{arccos}(z)=\dfrac{\pi}{2} + i\log\left( iz + \sqrt{1-z^2} \right),$$ where $i$ denotes the imaginary number and $\log$ denotes the logarithm.

Properties

Arccos as inverse cosine
Derivative of arccos
Antiderivative of arccos

References

Weisstein, Eric W. "Inverse Cosine." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/InverseCosine.html

See Also

Cosine
Cosh
Arccosh

Inverse trigonometric functions