Difference between revisions of "Spherical Bessel j"

From specialfunctionswiki
Jump to: navigation, search
Line 1: Line 1:
 
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">

Revision as of 20:10, 9 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

Theorem

The following formula holds for non-negative integers $n$: $$j_n(z)=(-1)^nz^n \left( \dfrac{1}{z} \dfrac{\mathrm{d}}{\mathrm{d}z} \right)^n \left( \dfrac{\sin z}{z} \right),$$ where $j_n$ denotes the spherical Bessel function of the first kind and $\sin$ denotes the sine function.

Proof

References

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:

<center>Bessel functions
</center>