Difference between revisions of "Derivative of Bessel J with respect to its order"
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(Created page with "==Theorem== The following formula holds: $$\dfrac{\partial}{\partial \nu} J_{\nu}(z)= J_{\nu}(z) \log \left( \dfrac{z}{2} \right) - z^{\nu} \displaystyle\sum_{k=0}^{\infty} (-...") |
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==References== | ==References== | ||
* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=findme|next=Derivative of Bessel Y with respect to its order}}: 9.1.64 | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=findme|next=Derivative of Bessel Y with respect to its order}}: 9.1.64 | ||
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| + | [[Category:Unproven]] | ||
Latest revision as of 22:45, 19 June 2016
Theorem
The following formula holds: $$\dfrac{\partial}{\partial \nu} J_{\nu}(z)= J_{\nu}(z) \log \left( \dfrac{z}{2} \right) - z^{\nu} \displaystyle\sum_{k=0}^{\infty} (-1)^k \dfrac{\psi(\nu+k+1)}{\Gamma(\nu+k+1)} \dfrac{z^{2k}}{k! 2^{2k+\nu}},$$ where $J_{\nu}$ denotes the Bessel function of the first kind, $\log$ denotes the logarithm, $\psi$ denotes the digamma function, and $k!$ denotes the factorial.
Proof
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 9.1.64
