Difference between revisions of "Spherical Bessel j"
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Revision as of 19:15, 10 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
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: █
References
Spherical Bessel $j_{\nu}$
