Difference between revisions of "Sine integral"

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The sine integral is defined by
 
The sine integral is defined by
$$\mathrm{Si}(z) = \displaystyle\int_0^z \mathrm{sinc}(t) \mathrm{d}t; |\mathrm{arg} z|<\pi,$$
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$$\mathrm{Si}(z) = \displaystyle\int_0^z \mathrm{sinc}(t) \mathrm{d}t, \quad |\mathrm{arg} \hspace{2pt} z|<\pi,$$
where $\mathrm{sinc}$ denotes the [[Sinc]] function.
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where $\mathrm{sinc}$ denotes the [[sinc]] function.
  
 
<div align="center">
 
<div align="center">
 
<gallery>
 
<gallery>
File:Si.png| Graph of $\mathrm{Si}$.
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File:Siplot.png| Graph of $\mathrm{Si}$.
File:Domain coloring sine integral.png|[[Domain coloring]] of [[analytic continuation]] of $\mathrm{Si}$.
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File:Complexsiplot.png|[[Domain coloring]] of $\mathrm{Si}$.
 
</gallery>
 
</gallery>
 
</div>
 
</div>
  
=Relationship to other functions=
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=Properties=
{{:Relationship between exponential integral Ei, cosine integral, and sine integral}}
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[[Derivative of sine integral]]<br />
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[[Antiderivative of sine integral]]<br />
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[[Relationship between exponential integral Ei, cosine integral, and sine integral]]<br />
  
 
=Videos=
 
=Videos=
[https://www.youtube.com/watch?v=hMW7aIYoN7U Laplace Transform of Sine Integral]
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[https://www.youtube.com/watch?v=hMW7aIYoN7U Laplace Transform of Sine Integral (2 January 2015)]
  
 
=References=
 
=References=
*[http://dlmf.nist.gov/8.21 Generalized Sine and Cosine Integrals]
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* {{BookReference|Special Functions of Mathematical Physics and Chemistry|1956|Ian N. Sneddon|prev=Cosine integral|next=Error function}}: $\S 5 (5.10)$
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* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=findme|next=}}: $5.2.1$
  
<center>{{:*-integral functions footer}}</center>
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{{:*-integral functions footer}}
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[[Category:SpecialFunction]]

Latest revision as of 00:42, 25 June 2017

The sine integral is defined by $$\mathrm{Si}(z) = \displaystyle\int_0^z \mathrm{sinc}(t) \mathrm{d}t, \quad |\mathrm{arg} \hspace{2pt} z|<\pi,$$ where $\mathrm{sinc}$ denotes the sinc function.

Properties

Derivative of sine integral
Antiderivative of sine integral
Relationship between exponential integral Ei, cosine integral, and sine integral

Videos

Laplace Transform of Sine Integral (2 January 2015)

References

$\ast$-integral functions