Difference between revisions of "Spherical Bessel y"

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Revision as of 18:36, 24 May 2016

The spherical Bessel function of the second kind is $$y_{\nu}(z)=\sqrt{\dfrac{\pi}{2z}} Y_{\nu+\frac{1}{2}}(z),$$ where $Y_{\nu}$ denotes the Bessel function of the second kind.

Properties

Theorem

The following formula holds for non-negative integers $n$: $$y_n(z)=(-1)^{n+1}z^n \left( \dfrac{1}{z} \dfrac{d}{dz} \right)^n \left( \dfrac{\cos z}{z} \right),$$ where $y_n$ denotes the spherical Bessel function of the second kind and $\cos$ denotes the cosine function.

Proof

References

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