# §10.15 Derivatives with Respect to Order

## Noninteger Values of $\nu$

 10.15.1 $\frac{\partial J_{\pm\nu}\left(z\right)}{\partial\nu}=\pm J_{\pm\nu}\left(z% \right)\ln\left(\tfrac{1}{2}z\right)\mp(\tfrac{1}{2}z)^{\pm\nu}\sum_{k=0}^{% \infty}(-1)^{k}\frac{\psi\left(k+1\pm\nu\right)}{\Gamma\left(k+1\pm\nu\right)}% \frac{(\tfrac{1}{4}z^{2})^{k}}{k!},$ ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$: gamma function, $\psi\left(\NVar{z}\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln\NVar{z}$: principal branch of logarithm function, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$: partial derivative, $\,\partial\NVar{x}$: partial differential, $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter A&S Ref: 9.1.64 Referenced by: §10.15, §10.38, Erratum (V1.0.17) for Equations (10.15.1), (10.38.1) Permalink: http://dlmf.nist.gov/10.15.E1 Encodings: TeX, pMML, png Addition (effective with 1.0.17): This equation has been generalized to include the additional case of $\ifrac{\partial J_{-\nu}\left(z\right)}{\partial\nu}$ because it will help the user who wants to combine it with (10.15.2). See also: Annotations for §10.15, §10.15 and Ch.10
 10.15.2 $\frac{\partial Y_{\nu}\left(z\right)}{\partial\nu}=\cot\left(\nu\pi\right)% \left(\frac{\partial J_{\nu}\left(z\right)}{\partial\nu}-\pi Y_{\nu}\left(z% \right)\right)-\csc\left(\nu\pi\right)\frac{\partial J_{-\nu}\left(z\right)}{% \partial\nu}-\pi J_{\nu}\left(z\right).$

## Integer Values of $\nu$

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

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

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

## Half-Integer Values of $\nu$

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

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

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