Difference between revisions of "Spherical Bessel j"
From specialfunctionswiki
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The spherical Bessel function of the first kind is defined by | The spherical Bessel function of the first kind is defined by | ||
$$j_{\nu}(z)=\sqrt{\dfrac{\pi}{2z}}J_{\nu + \frac{1}{2}}(z),$$ | $$j_{\nu}(z)=\sqrt{\dfrac{\pi}{2z}}J_{\nu + \frac{1}{2}}(z),$$ | ||
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=Properties= | =Properties= | ||
| − | + | [[Relationship between spherical Bessel j sub nu and sine]]<br /> | |
<div class="toccolours mw-collapsible mw-collapsed"> | <div class="toccolours mw-collapsible mw-collapsed"> | ||
Revision as of 22:44, 20 June 2016
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.
Domain coloring of $j_0$.
Properties
Relationship between spherical Bessel j sub nu and sine
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: █
References
Spherical Bessel $j_{\nu}$
