Difference between revisions of "Anger three-term recurrence"
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| − | + | ==Theorem== | |
| − | + | The following formula holds: | |
$$\textbf{J}_{\nu-1}(z)+\textbf{J}_{\nu+1}(z)=\dfrac{2\nu}{z}\textbf{J}_{\nu}(z)-\dfrac{2}{\pi z}\sin(\pi \nu),$$ | $$\textbf{J}_{\nu-1}(z)+\textbf{J}_{\nu+1}(z)=\dfrac{2\nu}{z}\textbf{J}_{\nu}(z)-\dfrac{2}{\pi z}\sin(\pi \nu),$$ | ||
where $\textbf{J}_{\nu}$ denote the [[Anger function]], $\pi$ denotes [[pi]], and $\sin$ denotes [[sine]]. | where $\textbf{J}_{\nu}$ denote the [[Anger function]], $\pi$ denotes [[pi]], and $\sin$ denotes [[sine]]. | ||
| − | + | ||
| − | + | ==Proof== | |
| − | + | ||
| − | + | ==References== | |
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| + | [[Category:Theorem]] | ||
Latest revision as of 05:50, 6 June 2016
Theorem
The following formula holds: $$\textbf{J}_{\nu-1}(z)+\textbf{J}_{\nu+1}(z)=\dfrac{2\nu}{z}\textbf{J}_{\nu}(z)-\dfrac{2}{\pi z}\sin(\pi \nu),$$ where $\textbf{J}_{\nu}$ denote the Anger function, $\pi$ denotes pi, and $\sin$ denotes sine.
