Difference between revisions of "Relationship between Bessel J and hypergeometric 0F1"
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The following formula holds: | 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|Bessel function of the first kind]], $\Gamma$ denotes the [[gamma]] function and ${}_0F_1$ denotes the [[hypergeometric | + | where $J_{\nu}$ denotes the [[Bessel J|Bessel function of the first kind]], $\Gamma$ denotes the [[gamma]] function and ${}_0F_1$ denotes the [[hypergeometric 0F1]]. |
==Proof== | ==Proof== | ||
Latest revision as of 06:00, 10 January 2017
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 0F1.
