Derivative of Bessel Y with respect to its order

Theorem

The following formula holds for $\nu \neq 0, \pm 1, \pm 2, \ldots$: $$\dfrac{\partial}{\partial \nu} Y_{\nu}(z)=\cot(\nu \pi) \left[ \dfrac{\partial}{\partial \nu} J_{\nu}(z)-\pi Y_{\nu}(z) \right] - \csc(\nu \pi) \dfrac{\partial}{\partial \nu} J_{-\nu}(z)-\pi J_{\nu}(z),$$ where $Y_{\nu}$ denotes the Bessel function of the second kind, $\cot$ denotes the cotangent, $J_{\nu}$ denotes the Bessel function of the first kind, $\pi$ denotes pi, and $\csc$ denotes the cosecant.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 9.1.65