Difference between revisions of "Spherical Bessel j"
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(Created page with "The spherical Bessel function of the first kind is defined by $$j_{\nu}(z)=\sqrt{\dfrac{\pi}{2z}}J_{\nu + \frac{1}{2}}(z),$$ where $J_{\nu}$ denotes the Bessel J sub nu|Bess...") |
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$$j_{\nu}(z)=\sqrt{\dfrac{\pi}{2z}}J_{\nu + \frac{1}{2}}(z),$$ | $$j_{\nu}(z)=\sqrt{\dfrac{\pi}{2z}}J_{\nu + \frac{1}{2}}(z),$$ | ||
where $J_{\nu}$ denotes the [[Bessel J sub nu|Bessel function of the first kind]]. | where $J_{\nu}$ denotes the [[Bessel J sub nu|Bessel function of the first kind]]. | ||
| + | |||
| + | =Properties= | ||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> The following formula holds: | ||
| + | $$1=\displaystyle\sum_{k=0}^{\infty} (2k+1)j_k(z)^2.$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
Revision as of 05:28, 16 May 2015
The spherical Bessel function of the first kind is defined by $$j_{\nu}(z)=\sqrt{\dfrac{\pi}{2z}}J_{\nu + \frac{1}{2}}(z),$$ where $J_{\nu}$ denotes the Bessel function of the first kind.
Properties
Theorem: The following formula holds: $$1=\displaystyle\sum_{k=0}^{\infty} (2k+1)j_k(z)^2.$$
Proof: █
