Difference between revisions of "Sine"
From specialfunctionswiki
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The sine function $\sin \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined by | The sine function $\sin \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined by | ||
$$\sin(z)=\dfrac{e^{iz}-e^{-iz}}{2i},$$ | $$\sin(z)=\dfrac{e^{iz}-e^{-iz}}{2i},$$ | ||
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</div> | </div> | ||
| − | =Properties= | + | ==Properties== |
[[Derivative of sine]]<br /> | [[Derivative of sine]]<br /> | ||
[[Pythagorean identity for sin and cos]]<br /> | [[Pythagorean identity for sin and cos]]<br /> | ||
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[[Relationship between tanh, inverse Gudermannian, and sin]]<br /> | [[Relationship between tanh, inverse Gudermannian, and sin]]<br /> | ||
| − | =Videos= | + | ==Videos== |
[https://www.youtube.com/watch?v=WD-n26cAFm0] | [https://www.youtube.com/watch?v=WD-n26cAFm0] | ||
| − | =See Also= | + | ==See Also== |
[[Arcsin]] <br /> | [[Arcsin]] <br /> | ||
[[Arcsinh]] <br /> | [[Arcsinh]] <br /> | ||
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[[Sinh]] <br /> | [[Sinh]] <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] | [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] | ||
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[[Category:SpecialFunction]] | [[Category:SpecialFunction]] | ||
| + | [[Category:Definition]] | ||
Revision as of 03:49, 6 June 2016
Definition
The sine function $\sin \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined by $$\sin(z)=\dfrac{e^{iz}-e^{-iz}}{2i},$$ where $e^z$ is the exponential function.
Graph of $\sin$ on $[-2\pi,2\pi]$.
Domain coloring of $\sin$.
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
See Also
References
The sine product formula and the gamma function
