Difference between revisions of "Relationship between spherical Bessel j and sine"

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==Theorem==
<strong>[[Relationship between spherical Bessel j sub nu and sine|Theorem]]:</strong> The following formula holds for non-negative integers $n$:
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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 sub nu|spherical Bessel function of the first kind]] and $\sin$ denotes the [[sine]] function.
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where $j_n$ denotes the [[Spherical Bessel j|spherical Bessel function of the first kind]] and $\sin$ denotes the [[sine]] function.
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<strong>Proof:</strong> █
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==Proof==
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==References==

Revision as of 00:39, 4 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.

Proof

References