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 pFq]].
+
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.

Proof

References