Difference between revisions of "Relationship between spherical Bessel j and sine"
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| − | + | ==Theorem== | |
| − | + | The following formula holds for non-negative integers $n$: | |
$$j_n(z)=(-1)^nz^n \left( \dfrac{1}{z} \dfrac{\mathrm{d}}{\mathrm{d}z} \right)^n \left( \dfrac{\sin z}{z} \right),$$ | $$j_n(z)=(-1)^nz^n \left( \dfrac{1}{z} \dfrac{\mathrm{d}}{\mathrm{d}z} \right)^n \left( \dfrac{\sin z}{z} \right),$$ | ||
| − | where $j_n$ denotes the [[Spherical Bessel j | + | where $j_n$ denotes the [[Spherical Bessel j|spherical Bessel function of the first kind]] and $\sin$ denotes the [[sine]] function. |
| − | + | ||
| − | + | ==Proof== | |
| − | + | ||
| − | + | ==References== | |
| + | |||
| + | [[Category:Theorem]] | ||
| + | [[Category:Unproven]] | ||
Latest revision as of 07:34, 8 June 2016
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.
