Difference between revisions of "Relationship between spherical Bessel y and cosine"
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| − | + | ==Theorem== | |
| − | + | The following formula holds for non-negative integers $n$: | |
$$y_n(z)=(-1)^{n+1}z^n \left( \dfrac{1}{z} \dfrac{d}{dz} \right)^n \left( \dfrac{\cos z}{z} \right),$$ | $$y_n(z)=(-1)^{n+1}z^n \left( \dfrac{1}{z} \dfrac{d}{dz} \right)^n \left( \dfrac{\cos z}{z} \right),$$ | ||
where $y_n$ denotes the [[Spherical Bessel y sub nu|spherical Bessel function of the second kind]] and $\cos$ denotes the [[cosine]] function. | where $y_n$ denotes the [[Spherical Bessel y sub nu|spherical Bessel function of the second kind]] and $\cos$ denotes the [[cosine]] function. | ||
| − | + | ||
| − | + | ==Proof== | |
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| − | + | ==References== | |
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| + | [[Category:Theorem]] | ||
| + | [[Category:Unproven]] | ||
Revision as of 07:40, 8 June 2016
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.
