Difference between revisions of "Spherical Bessel j"
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| + | <strong>Theorem:</strong> The following formula holds for non-negative integers $n$: | ||
| + | $$j_n(z)=(-1)^nz^n \left( \dfrac{1}{z} \dfrac{d}{dz} \right)^n \left( \dfrac{\sin z}{z} \right).$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
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<div class="toccolours mw-collapsible mw-collapsed"> | <div class="toccolours mw-collapsible mw-collapsed"> | ||
<strong>Proposition:</strong> The following formula holds: | <strong>Proposition:</strong> The following formula holds: | ||
Revision as of 06:33, 10 June 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 for non-negative integers $n$: $$j_n(z)=(-1)^nz^n \left( \dfrac{1}{z} \dfrac{d}{dz} \right)^n \left( \dfrac{\sin z}{z} \right).$$
Proof: █
Proposition: The following formula holds: $$1=\displaystyle\sum_{k=0}^{\infty} (2k+1)j_k(z)^2.$$
Proof: █
Proposition: The following formula holds: $$\dfrac{\sin(2z)}{2z} = \displaystyle\sum_{k=0}^{\infty} (-1)^k(2k+1)j_k(z)^2.$$
Proof: █
Spherical Bessel $j_{\nu}$
