# §10.15 Derivatives with Respect to Order

## Noninteger Values of $\nu$

 10.15.1 $\frac{\partial\mathop{J_{\nu}\/}\nolimits\!\left(z\right)}{\partial\nu}=% \mathop{J_{\nu}\/}\nolimits\!\left(z\right)\mathop{\ln\/}\nolimits\!\left(% \tfrac{1}{2}z\right)-(\tfrac{1}{2}z)^{\nu}\sum_{k=0}^{\infty}(-1)^{k}\frac{% \mathop{\psi\/}\nolimits\!\left(\nu+k+1\right)}{\mathop{\Gamma\/}\nolimits\!% \left(\nu+k+1\right)}\frac{(\tfrac{1}{4}z^{2})^{k}}{k!},$
 10.15.2 $\frac{\partial\mathop{Y_{\nu}\/}\nolimits\!\left(z\right)}{\partial\nu}=% \mathop{\cot\/}\nolimits\!\left(\nu\pi\right)\left(\frac{\partial\mathop{J_{% \nu}\/}\nolimits\!\left(z\right)}{\partial\nu}-\pi\mathop{Y_{\nu}\/}\nolimits% \!\left(z\right)\right)-\mathop{\csc\/}\nolimits\!\left(\nu\pi\right)\frac{% \partial\mathop{J_{-\nu}\/}\nolimits\!\left(z\right)}{\partial\nu}-\pi\mathop{% J_{\nu}\/}\nolimits\!\left(z\right).$

## Integer Values of $\nu$

 10.15.3 $\left.\frac{\partial\mathop{J_{\nu}\/}\nolimits\!\left(z\right)}{\partial\nu}% \right|_{\nu=n}=\frac{\pi}{2}\mathop{Y_{n}\/}\nolimits\!\left(z\right)+\frac{n% !}{2(\tfrac{1}{2}z)^{n}}\sum_{k=0}^{n-1}\frac{(\tfrac{1}{2}z)^{k}\mathop{J_{k}% \/}\nolimits\!\left(z\right)}{k!(n-k)}.$

For $\ifrac{\partial\mathop{J_{\nu}\/}\nolimits\!\left(z\right)}{\partial\nu}$ at $\nu=-n$ combine (10.2.4) and (10.15.3).

 10.15.4 $\displaystyle\left.\frac{\partial\mathop{Y_{\nu}\/}\nolimits\!\left(z\right)}{% \partial\nu}\right|_{\nu=n}$ $\displaystyle=-\frac{\pi}{2}\mathop{J_{n}\/}\nolimits\!\left(z\right)+\frac{n!% }{2(\tfrac{1}{2}z)^{n}}\sum_{k=0}^{n-1}\frac{(\tfrac{1}{2}z)^{k}\mathop{Y_{k}% \/}\nolimits\!\left(z\right)}{k!(n-k)},$ 10.15.5 $\displaystyle\left.\frac{\partial\mathop{J_{\nu}\/}\nolimits\!\left(z\right)}{% \partial\nu}\right|_{\nu=0}$ $\displaystyle=\frac{\pi}{2}\mathop{Y_{0}\/}\nolimits\!\left(z\right),\quad% \left.\frac{\partial\mathop{Y_{\nu}\/}\nolimits\!\left(z\right)}{\partial\nu}% \right|_{\nu=0}=-\frac{\pi}{2}\mathop{J_{0}\/}\nolimits\!\left(z\right).$

## Half-Integer Values of $\nu$

For the notations $\mathop{\mathrm{Ci}\/}\nolimits$ and $\mathop{\mathrm{Si}\/}\nolimits$ see §6.2(ii). When $x>0$,

 10.15.6 $\displaystyle\left.\frac{\partial\mathop{J_{\nu}\/}\nolimits\!\left(x\right)}{% \partial\nu}\right|_{\nu=\frac{1}{2}}$ $\displaystyle=\sqrt{\frac{2}{\pi x}}\left(\mathop{\mathrm{Ci}\/}\nolimits\!% \left(2x\right)\mathop{\sin\/}\nolimits x-\mathop{\mathrm{Si}\/}\nolimits\!% \left(2x\right)\mathop{\cos\/}\nolimits x\right),$ 10.15.7 $\displaystyle\left.\frac{\partial\mathop{J_{\nu}\/}\nolimits\!\left(x\right)}{% \partial\nu}\right|_{\nu=-\frac{1}{2}}$ $\displaystyle=\sqrt{\frac{2}{\pi x}}\left(\mathop{\mathrm{Ci}\/}\nolimits\!% \left(2x\right)\mathop{\cos\/}\nolimits x+\mathop{\mathrm{Si}\/}\nolimits\!% \left(2x\right)\mathop{\sin\/}\nolimits x\right),$ 10.15.8 $\displaystyle\left.\frac{\partial\mathop{Y_{\nu}\/}\nolimits\!\left(x\right)}{% \partial\nu}\right|_{\nu=\frac{1}{2}}$ $\displaystyle=\sqrt{\frac{2}{\pi x}}\left(\mathop{\mathrm{Ci}\/}\nolimits\!% \left(2x\right)\mathop{\cos\/}\nolimits x+\left(\mathop{\mathrm{Si}\/}% \nolimits\!\left(2x\right)-\pi\right)\mathop{\sin\/}\nolimits x\right),$ 10.15.9 $\displaystyle\left.\frac{\partial\mathop{Y_{\nu}\/}\nolimits\!\left(x\right)}{% \partial\nu}\right|_{\nu=-\frac{1}{2}}$ $\displaystyle=-\sqrt{\frac{2}{\pi x}}\left(\mathop{\mathrm{Ci}\/}\nolimits\!% \left(2x\right)\mathop{\sin\/}\nolimits x-\left(\mathop{\mathrm{Si}\/}% \nolimits\!\left(2x\right)-\pi\right)\mathop{\cos\/}\nolimits x\right).$

For further results see Brychkov and Geddes (2005) and Landau (1999, 2000).