Difference between revisions of "Spherical Bessel j"

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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
 
$$j_{\nu}(z)=\sqrt{\dfrac{\pi}{2z}}J_{\nu + \frac{1}{2}}(z),$$
 
$$j_{\nu}(z)=\sqrt{\dfrac{\pi}{2z}}J_{\nu + \frac{1}{2}}(z),$$
where $J_{\nu}$ denotes the [[Bessel J sub nu|Bessel function of the first kind]].
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where $J_{\nu}$ denotes the [[Bessel J|Bessel function of the first kind]].
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File:Domcolsphericalbesseljsub0.png|[[Domain coloring]] of $j_0$.
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=Properties=
 
=Properties=
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[[Relationship between spherical Bessel j sub nu and sine]]<br />
<strong>Theorem:</strong> The following formula holds for non-negative integers $n$:
 
$$j_n(z)=(-1)^nz^n \left( \dfrac{1}{z} \dfrac{d}{dz} \right)^n \left( \dfrac{\sin z}{z} \right).$$
 
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<strong>Proof:</strong> █
 
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=References=
<strong>Proposition:</strong> The following formula holds:
 
$$1=\displaystyle\sum_{k=0}^{\infty} (2k+1)j_k(z)^2.$$
 
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<strong>Proof:</strong> █
 
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{{:Bessel functions footer}}
<strong>Proposition:</strong> The following formula holds:
 
$$\dfrac{\sin(2z)}{2z} = \displaystyle\sum_{k=0}^{\infty} (-1)^k(2k+1)j_k(z)^2.$$
 
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<strong>Proof:</strong> █
 
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<center>{{:Bessel functions footer}}</center>
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[[Category:SpecialFunction]]

Latest 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.

Properties

Relationship between spherical Bessel j sub nu and sine

References

Bessel functions