Difference between revisions of "Sine"

From specialfunctionswiki
Jump to: navigation, search
(Properties)
 
(37 intermediate revisions by the same user not shown)
Line 1: Line 1:
The sine function $\sin \colon \mathbb{R} \rightarrow \mathbb{R}$ is the unique solution of the second order initial value problem $y''=-y;y(0)=0,y'(0)=1$.
+
__NOTOC__
 +
The sine function $\sin \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined by
 +
$$\sin(z)=\dfrac{e^{iz}-e^{-iz}}{2i},$$
 +
where $e^{iz}$ denotes the [[exponential]].
  
 
<div align="center">
 
<div align="center">
 
<gallery>
 
<gallery>
File:Sine.png|Graph of $\sin$ on $\mathbb{R}$.
+
File:Sineplot.png|Graph of $\sin$ on $[-2\pi,2\pi]$.
File:Complex sin.jpg|[[Domain coloring]] of [[analytic continuation]] of $\sin$.
+
File:Complexsineplot.png|[[Domain coloring]] of $\sin$.
 +
File:Trig Functions Diagram.svg|Trigonometric functions diagram using the unit circle.
 
</gallery>
 
</gallery>
 
</div>
 
</div>
  
 
=Properties=
 
=Properties=
{{:Derivative of sine}}
+
[[Derivative of sine]]<br />
 +
[[Pythagorean identity for sin and cos]]<br />
 +
[[Taylor series of sine]]<br />
 +
[[Weierstrass factorization of sine]]<br />
 +
[[Euler's reflection formula for gamma]]<br />
 +
[[Beta in terms of sine and cosine]]<br />
 +
[[Relationship between sine and hypergeometric 0F1]]<br />
 +
[[Relationship between spherical Bessel j sub nu and sine]]<br />
 +
[[Relationship between sin and sinh]]<br />
 +
[[Relationship between sinh and sin]]<br />
 +
[[Relationship between sine, Gudermannian, and tanh]]<br />
 +
[[Relationship between tanh, inverse Gudermannian, and sin]]<br />
  
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
+
=Videos=
<strong>Proposition:</strong> $\sin(x)=\displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^kx^{2k+1}}{(2k+1)!}$
+
[https://www.youtube.com/watch?v=WD-n26cAFm0]
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> proof goes here █
 
</div>
 
</div>
 
  
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
+
=See Also=
<strong>Proposition:</strong> $\sin(x) = x \displaystyle\prod_{k=1}^{\infty} \left( 1 - \dfrac{x^2}{k^2\pi^2} \right)$
+
[[Arcsin]] <br />
<div class="mw-collapsible-content">
+
[[Arcsinh]] <br />
<strong>Proof:</strong> proof goes here █
+
[[Cosecant]]<br />
</div>
+
[[Sinh]] <br />
</div>
 
  
{{:Gamma-Sine_Relation}}
+
=External links=
 +
[http://ocw.mit.edu/courses/mathematics/18-104-seminar-in-analysis-applications-to-number-theory-fall-2006/projects/chan.pdf The sine product formula and the gamma function]<br />
  
 
=References=
 
=References=
[http://ocw.mit.edu/courses/mathematics/18-104-seminar-in-analysis-applications-to-number-theory-fall-2006/projects/chan.pdf The sine product formula and the gamma function]
+
* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=findme|next=Cosine}}: $4.3.1$
 +
 
 +
{{:Trigonometric functions footer}}
 +
 
 +
[[Category:SpecialFunction]]
 +
[[Category:Definition]]

Latest revision as of 17:34, 1 July 2017

The sine function $\sin \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined by $$\sin(z)=\dfrac{e^{iz}-e^{-iz}}{2i},$$ where $e^{iz}$ denotes the exponential.

Properties

Derivative of sine
Pythagorean identity for sin and cos
Taylor series of sine
Weierstrass factorization of sine
Euler's reflection formula for gamma
Beta in terms of sine and cosine
Relationship between sine and hypergeometric 0F1
Relationship between spherical Bessel j sub nu and sine
Relationship between sin and sinh
Relationship between sinh and sin
Relationship between sine, Gudermannian, and tanh
Relationship between tanh, inverse Gudermannian, and sin

Videos

[1]

See Also

Arcsin
Arcsinh
Cosecant
Sinh

External links

The sine product formula and the gamma function

References

Trigonometric functions