Derivative of Bessel J with respect to its order

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.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 9.1.64