Difference between revisions of "Spherical Bessel j"

From specialfunctionswiki
Jump to: navigation, search
 
(5 intermediate revisions by the same user not shown)
Line 1: Line 1:
 +
__NOTOC__
 
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]].
+
where $J_{\nu}$ denotes the [[Bessel J|Bessel function of the first kind]].
  
 
<div align="center">
 
<div align="center">
Line 10: Line 11:
  
 
=Properties=
 
=Properties=
{{:Relationship between spherical Bessel j sub nu and sine}}
+
[[Relationship between spherical Bessel j sub nu and sine]]<br />
  
<div class="toccolours mw-collapsible mw-collapsed">
+
=References=
<strong>Proposition:</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>
 
  
<div class="toccolours mw-collapsible mw-collapsed">
+
{{:Bessel functions footer}}
<strong>Proposition:</strong> The following formula holds:
 
$$\dfrac{\sin(2z)}{2z} = \displaystyle\sum_{k=0}^{\infty} (-1)^k(2k+1)j_k(z)^2.$$
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
</div>
 
</div>
 
  
<center>{{:Bessel functions footer}}</center>
+
[[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