Difference between revisions of "Anger function"
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| − | The Anger function is defined by | + | [[Category:SpecialFunction]] |
| + | 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 $\mathbf{J}_{\frac{1}{2}}$. | ||
| + | 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:Complexanger0plot.png|[[Domain coloring]] of $\mathbf{J}_0$. | File:Complexanger0plot.png|[[Domain coloring]] of $\mathbf{J}_0$. | ||
| − | File: | + | File:Complexanger0.5plot.png|[[Domain coloring]] of $\mathbf{J}_{\frac{1}{2}}$. |
| + | File:Complexanger1plot.png|[[Domain coloring]] of $\mathbf{J}_1$. | ||
| + | File:Complexanger1.5plot.png|[[Domain coloring]] of $\mathbf{J}_{\frac{3}{2}}$. | ||
</gallery> | </gallery> | ||
</div> | </div> | ||
=Properties= | =Properties= | ||
| − | + | [[Value of Anger at 0]]<br /> | |
| − | + | [[Anger recurrence relation]]<br /> | |
| − | + | [[Anger derivative recurrence]]<br /> | |
| − | + | [[Relationship between Anger function and Bessel J sub nu]]<br /> | |
| − | + | [[Relationship between Weber function and Anger function]]<br /> | |
| − | + | [[Relationship between Anger function and Weber function]]<br /> | |
=See Also= | =See Also= | ||
[[Bessel J]]<br /> | [[Bessel J]]<br /> | ||
| − | [[Weber function]] | + | [[Weber function]]<br /> |
=References= | =References= | ||
| − | + | * {{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 $$\mathbf{J}_{\nu}(z) = \dfrac{1}{\pi} \displaystyle\int_0^{\pi} \cos(\nu \theta - z \sin(\theta)) \mathrm{d}\theta.$$
Graph of $\mathbf{J}_0$.
Graph of $\mathbf{J}_{\frac{1}{2}}$.
Graph of $\mathbf{J}_{\frac{3}{2}}$.
Graph of $\mathbf{J}_2$.
Domain coloring of $\mathbf{J}_0$.
Domain coloring of $\mathbf{J}_{\frac{1}{2}}$.
Domain coloring of $\mathbf{J}_1$.
Domain coloring of $\mathbf{J}_{\frac{3}{2}}$.
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
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 12.3.1
