Difference between revisions of "Relationship between spherical Bessel j and sine"
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<strong>[[Relationship between spherical Bessel j sub nu and sine|Theorem]]:</strong> The following formula holds for non-negative integers $n$: | <strong>[[Relationship between spherical Bessel j sub nu and sine|Theorem]]:</strong> The following formula holds for non-negative integers $n$: | ||
| − | $$j_n(z)=(-1)^nz^n \left( \dfrac{1}{z} \dfrac{d}{ | + | $$j_n(z)=(-1)^nz^n \left( \dfrac{1}{z} \dfrac{\mathrm{d}}{\mathrm{d}z} \right)^n \left( \dfrac{\sin z}{z} \right),$$ |
where $j_n$ denotes the [[Spherical Bessel j sub nu|spherical Bessel function of the first kind]] and $\sin$ denotes the [[sine]] function. | where $j_n$ denotes the [[Spherical Bessel j sub nu|spherical Bessel function of the first kind]] and $\sin$ denotes the [[sine]] function. | ||
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Revision as of 20:16, 15 May 2016
Theorem: The following formula holds for non-negative integers $n$: $$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.
Proof: █
