Difference between revisions of "Anger function"

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The Anger function is defined by
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[[Category:SpecialFunction]]
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Let $\nu \in \mathbb{C}$. The Anger function $\mathbf{J}_{\nu}$ is defined by
 
$$\mathbf{J}_{\nu}(z) = \dfrac{1}{\pi} \displaystyle\int_0^{\pi} \cos(\nu \theta - z \sin(\theta)) \mathrm{d}\theta.$$
 
$$\mathbf{J}_{\nu}(z) = \dfrac{1}{\pi} \displaystyle\int_0^{\pi} \cos(\nu \theta - z \sin(\theta)) \mathrm{d}\theta.$$
  
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File:Anger0plot.png | Graph of $\mathbf{J}_0$.
 
File:Anger0plot.png | Graph of $\mathbf{J}_0$.
 
File:Anger0.5plot.png|Graph of $\mathbf{J}_{\frac{1}{2}}$.
 
File:Anger0.5plot.png|Graph of $\mathbf{J}_{\frac{1}{2}}$.
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File:Anger1.5plot.png|Graph of $\mathbf{J}_{\frac{3}{2}}$.
 
File:Anger2plot.png|Graph of $\mathbf{J}_2$.
 
File:Anger2plot.png|Graph of $\mathbf{J}_2$.
File:Domaincoloring angerj.png|[[Domain coloring]] of [[analytic continuation]] of $\mathbf{J}_{\frac{1}{2}}$.
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File:Complexanger0plot.png|[[Domain coloring]] of $\mathbf{J}_0$.
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File:Complexanger0.5plot.png|[[Domain coloring]] of $\mathbf{J}_{\frac{1}{2}}$.
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File:Complexanger1plot.png|[[Domain coloring]] of $\mathbf{J}_1$.
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File:Complexanger1.5plot.png|[[Domain coloring]] of $\mathbf{J}_{\frac{3}{2}}$.
 
</gallery>
 
</gallery>
 
</div>
 
</div>
  
 
=Properties=
 
=Properties=
{{:Value of Anger at 0}}
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[[Value of Anger at 0]]<br />
{{:Relationship between Anger function and Bessel J sub nu}}
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[[Anger recurrence relation]]<br />
{{:Relationship between Weber function and Anger function}}
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[[Anger derivative recurrence]]<br />
{{:Relationship between Anger function and Weber function}}
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[[Relationship between Anger function and Bessel J sub nu]]<br />
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[[Relationship between Weber function and Anger function]]<br />
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[[Relationship between Anger function and Weber function]]<br />
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=See Also=
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[[Bessel J]]<br />
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[[Weber function]]<br />
  
 
=References=
 
=References=
[http://dualaud.net/specialfunctionswiki/abramowitz_and_stegun-1.03/page_498.htm Abramowitz and Stegun]<br />
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* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Relationship between modified Struve L and modified spherical Bessel j functions|next=Anger of integer order is Bessel J}}: 12.3.1

Latest revision as of 04:05, 6 June 2016

Let $\nu \in \mathbb{C}$. The Anger function $\mathbf{J}_{\nu}$ is defined by $$\mathbf{J}_{\nu}(z) = \dfrac{1}{\pi} \displaystyle\int_0^{\pi} \cos(\nu \theta - z \sin(\theta)) \mathrm{d}\theta.$$

Properties

Value of Anger at 0
Anger recurrence relation
Anger derivative recurrence
Relationship between Anger function and Bessel J sub nu
Relationship between Weber function and Anger function
Relationship between Anger function and Weber function

See Also

Bessel J
Weber function

References