Difference between revisions of "Spherical Bessel j"

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=Properties=
 
=Properties=
 
[[Relationship between spherical Bessel j sub nu and sine]]<br />
 
[[Relationship between spherical Bessel j sub nu and sine]]<br />
 
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<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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<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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=References=
 
=References=

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