Spherical Bessel j

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 $n$: $$j_n(z)=(-1)^nz^n \left( \dfrac{1}{z} \dfrac{d}{dz} \right)^n \left( \dfrac{\sin z}{z} \right).$$

Proof: █

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

Anger function

Bessel-Clifford

Bessel $J_{\nu}$

Bessel $Y_{\nu}$

Bessel $I_{\nu}$

Modified Bessel $K_{\nu}$

Spherical Bessel $j_{\nu}$

Spherical Bessel $y_{\nu}$

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