Difference between revisions of "Arcsin"
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| − | The function $\mathrm{arcsin} \colon | + | __NOTOC__ |
| + | The function $\mathrm{arcsin} \colon \mathbb{C} \setminus \left\{ (-\infty,-1) \bigcup (1,\infty) \right\} \rightarrow \mathbb{C}$ is defined by | ||
| + | $$\rm{arcsin}(z)=-i \log \left( iz + \sqrt{1-z^2} \right),$$ | ||
| + | where $i$ denotes the [[imaginary number]] and $\log$ denotes the [[logarithm]]. <br /> | ||
<div align="center"> | <div align="center"> | ||
<gallery> | <gallery> | ||
| − | File: | + | File:Arcsinplot.png|Graph of $\mathrm{arcsin}$ on $[-1,1]$. |
| − | File: | + | File:Complexarcsinplot.png|[[Domain coloring]] of $\mathrm{arcsin}$. |
</gallery> | </gallery> | ||
</div> | </div> | ||
=Properties= | =Properties= | ||
| − | < | + | [[Arcsin as inverse sine]]<br /> |
| − | < | + | [[Derivative of arcsin]]<br /> |
| − | + | [[Antiderivative of arcsin]] <br /> | |
| − | < | + | [[Relationship between arcsin and arccsc]] <br /> |
| − | < | + | [[2F1(1/2,1/2;3/2;z^2)=arcsin(z)/z]]<br /> |
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| − | </ | ||
| − | + | =Videos= | |
| − | + | [https://www.youtube.com/watch?v=JGU74wbZMLg Inverse Trig Functions: Arcsin (1 October 2009)]<br /> | |
| − | + | [https://www.youtube.com/watch?v=JZ9Ku1TTeA4 What is arcsin(x)? (18 August 2011)]<br /> | |
| − | < | + | [https://www.youtube.com/watch?v=KmHD7CsOw5Y Integrate x*arcsin(x) (25 February 2013)]<br /> |
| − | + | [https://www.youtube.com/watch?v=4CY7RIUhs2s What is the inverse of arcsin(ln(x))? (28 April 2014)]<br /> | |
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| − | </ | ||
| − | + | =See Also= | |
| − | + | [[Sine]] <br /> | |
| − | + | [[Sinh]] <br /> | |
| − | + | [[Arcsinh]] | |
| − | < | ||
| − | </ | ||
| − | |||
| − | + | =References= | |
| − | + | *[http://mathworld.wolfram.com/InverseSine.html Weisstein, Eric W. "Inverse Sine." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/InverseSine.html]<br /> | |
| − | + | [http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=PPN600494829_0015%7CLOG_0028 On the function arc sin(x+iy)-Cayley]<br /> | |
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| − | </ | ||
| − | + | {{:Inverse trigonometric functions footer}} | |
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| − | + | [[Category:SpecialFunction]] | |
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Latest revision as of 23:45, 22 December 2016
The function $\mathrm{arcsin} \colon \mathbb{C} \setminus \left\{ (-\infty,-1) \bigcup (1,\infty) \right\} \rightarrow \mathbb{C}$ is defined by
$$\rm{arcsin}(z)=-i \log \left( iz + \sqrt{1-z^2} \right),$$
where $i$ denotes the imaginary number and $\log$ denotes the logarithm.
Graph of $\mathrm{arcsin}$ on $[-1,1]$.
Domain coloring of $\mathrm{arcsin}$.
Properties
Arcsin as inverse sine
Derivative of arcsin
Antiderivative of arcsin
Relationship between arcsin and arccsc
2F1(1/2,1/2;3/2;z^2)=arcsin(z)/z
Videos
Inverse Trig Functions: Arcsin (1 October 2009)
What is arcsin(x)? (18 August 2011)
Integrate x*arcsin(x) (25 February 2013)
What is the inverse of arcsin(ln(x))? (28 April 2014)
See Also
References
On the function arc sin(x+iy)-Cayley
