Difference between revisions of "Bessel Y"

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Bessel functions (of the second kind) $Y_{\nu}$ are defined via the formula
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Bessel functions of the second kind, $Y_{\nu}$, are defined via the formula
 
$$Y_{\nu}(z)=\dfrac{J_{\nu}(z)\cos(\nu \pi)-J_{-\nu}(z)}{\sin(\nu \pi)}.$$
 
$$Y_{\nu}(z)=\dfrac{J_{\nu}(z)\cos(\nu \pi)-J_{-\nu}(z)}{\sin(\nu \pi)}.$$
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Sometimes these functions are called Neumann functions and have the notation $N_{\nu}$ instead of $Y_{\nu}$.
  
 
<div align="center">
 
<div align="center">
 
<gallery>
 
<gallery>
File:Bessel y plot.png|Graph of $Y_0,Y_1,\ldots,Y_5$ on $[0,20]$.
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File:Bessely,n=0plot.png|Graph of $Y_0$.
File:Complex bessel y sub 0.png|[[Domain coloring]] of [[analytic continuation]] of $Y_0(z)$.
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File:Multiplebesselyplot.png|Graph of $Y_0,Y_1,Y_2$, and $Y_3$.
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File:Complexbessely,n=0.png|[[Domain coloring]] of $Y_0$.
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File:Complexbessely,n=1.png|[[Domain coloring]] of $Y_1$.
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File:Page 359Abramowitz-Stegun(Bessel functions).jpg|Bessel functions from [http://dualaud.net/specialfunctionswiki/abramowitz_and_stegun-1.03/ Abramowitz&Stegun]
 
</gallery>
 
</gallery>
 
</div>
 
</div>
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=Properties=
 
=Properties=
{{:Bessel J sub nu and Y sub nu solve Bessel's differential equation}}
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[[Derivative of Bessel Y with respect to its order]]
  
<center>{{:Bessel functions footer}}</center>
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=References=
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* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=findme|next=Hankel H (1)}}: 9.1.2
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[http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=PPN600494829_0021%7CLOG_0023 Bessel's functions of the second order - C.V. Coates]<br />
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{{:Bessel functions footer}}
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[[Category:SpecialFunction]]

Latest revision as of 15:39, 10 July 2017

Bessel functions of the second kind, $Y_{\nu}$, are defined via the formula $$Y_{\nu}(z)=\dfrac{J_{\nu}(z)\cos(\nu \pi)-J_{-\nu}(z)}{\sin(\nu \pi)}.$$ Sometimes these functions are called Neumann functions and have the notation $N_{\nu}$ instead of $Y_{\nu}$.


Properties

Derivative of Bessel Y with respect to its order

References

Bessel's functions of the second order - C.V. Coates

Bessel functions