Difference between revisions of "Relationship between Bessel J and hypergeometric 0F1"
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<strong>[[Relationship between Bessel J sub nu and hypergeometric 0F1|Theorem]]:</strong> The following formula holds: | <strong>[[Relationship between Bessel J sub nu and hypergeometric 0F1|Theorem]]:</strong> 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 [[ | + | 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]]. |
<div class="mw-collapsible-content"> | <div class="mw-collapsible-content"> | ||
<strong>Proof:</strong> █ | <strong>Proof:</strong> █ | ||
</div> | </div> | ||
</div> | </div> | ||
Revision as of 18:36, 20 May 2015
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.
Proof: █
