Difference between revisions of "Spherical Bessel y"
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$$y_{\nu}(z)=\sqrt{\dfrac{\pi}{2z}} Y_{\nu+\frac{1}{2}}(z),$$ | $$y_{\nu}(z)=\sqrt{\dfrac{\pi}{2z}} Y_{\nu+\frac{1}{2}}(z),$$ | ||
where $Y_{\nu}$ denotes the [[Bessel Y sub nu|Bessel function of the second kind]]. | where $Y_{\nu}$ denotes the [[Bessel Y sub nu|Bessel function of the second kind]]. | ||
| + | |||
| + | =Properties= | ||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> The following formula holds for non-negative integers $n$: | ||
| + | $$y_n(z)=(-1)^{n+1}z^n \left( \dfrac{1}{z} \dfrac{d}{dz} \right)^n \left( \dfrac{\cos z}{z} \right).$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
<center>{{:Bessel functions footer}}</center> | <center>{{:Bessel functions footer}}</center> | ||
Revision as of 06:35, 10 June 2015
The spherical Bessel function of the second kind is $$y_{\nu}(z)=\sqrt{\dfrac{\pi}{2z}} Y_{\nu+\frac{1}{2}}(z),$$ where $Y_{\nu}$ denotes the Bessel function of the second kind.
Properties
Theorem: The following formula holds for non-negative integers $n$: $$y_n(z)=(-1)^{n+1}z^n \left( \dfrac{1}{z} \dfrac{d}{dz} \right)^n \left( \dfrac{\cos z}{z} \right).$$
Proof: █
Spherical Bessel $y_{\nu}$
