Difference between revisions of "Relationship between Bessel J and hypergeometric 0F1"
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| − | + | ==Theorem== | |
| − | + | The following formula holds: | |
$$J_{\nu}(z) = \left( \dfrac{z}{2} \right)^{\nu} \dfrac{1}{\Gamma(\nu+1)} {}_0F_1 \left(-;\nu+1;-\dfrac{z^2}{4} \right),$$ | $$J_{\nu}(z) = \left( \dfrac{z}{2} \right)^{\nu} \dfrac{1}{\Gamma(\nu+1)} {}_0F_1 \left(-;\nu+1;-\dfrac{z^2}{4} \right),$$ | ||
where $J_{\nu}$ denotes the [[Bessel J sub nu|Bessel function of the first kind]], $\Gamma$ denotes the [[gamma]] function and ${}_0F_1$ denotes the [[hypergeometric pFq]]. | where $J_{\nu}$ denotes the [[Bessel J sub nu|Bessel function of the first kind]], $\Gamma$ denotes the [[gamma]] function and ${}_0F_1$ denotes the [[hypergeometric pFq]]. | ||
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| − | + | ==Proof== | |
| − | + | ||
| − | + | ==References== | |
Revision as of 19:57, 9 June 2016
Theorem
The following formula holds: $$J_{\nu}(z) = \left( \dfrac{z}{2} \right)^{\nu} \dfrac{1}{\Gamma(\nu+1)} {}_0F_1 \left(-;\nu+1;-\dfrac{z^2}{4} \right),$$ where $J_{\nu}$ denotes the Bessel function of the first kind, $\Gamma$ denotes the gamma function and ${}_0F_1$ denotes the hypergeometric pFq.
