§10.38 Derivatives with Respect to Order

 10.38.1 $\frac{\partial I_{\nu}\left(z\right)}{\partial\nu}=I_{\nu}\left(z\right)\ln% \left(\tfrac{1}{2}z\right)-(\tfrac{1}{2}z)^{\nu}\sum_{k=0}^{\infty}\frac{\psi% \left(\nu+k+1\right)}{\Gamma\left(\nu+k+1\right)}\frac{(\frac{1}{4}z^{2})^{k}}% {k!},$
 10.38.2 $\frac{\partial K_{\nu}\left(z\right)}{\partial\nu}=\tfrac{1}{2}\pi\csc\left(% \nu\pi\right)\*\left(\frac{\partial I_{-\nu}\left(z\right)}{\partial\nu}-\frac% {\partial I_{\nu}\left(z\right)}{\partial\nu}\right)-\pi\cot\left(\nu\pi\right% )K_{\nu}\left(z\right),$ $\nu\notin\mathbb{Z}$.

Integer Values of $\nu$

 10.38.3 $\left.(-1)^{n}\frac{\partial I_{\nu}\left(z\right)}{\partial\nu}\right|_{\nu=n% }=-K_{n}\left(z\right)+\frac{n!}{2(\frac{1}{2}z)^{n}}\sum_{k=0}^{n-1}(-1)^{k}% \frac{(\frac{1}{2}z)^{k}I_{k}\left(z\right)}{k!(n-k)},$

For $\ifrac{\partial I_{\nu}\left(z\right)}{\partial\nu}$ at $\nu=-n$ combine (10.38.1), (10.38.2), and (10.38.4).

 10.38.4 $\left.\frac{\partial K_{\nu}\left(z\right)}{\partial\nu}\right|_{\nu=n}=\frac{% n!}{2(\frac{1}{2}z)^{n}}\sum_{k=0}^{n-1}\frac{(\frac{1}{2}z)^{k}K_{k}\left(z% \right)}{k!(n-k)}.$
 10.38.5 $\displaystyle\left.\frac{\partial I_{\nu}\left(z\right)}{\partial\nu}\right|_{% \nu=0}$ $\displaystyle=-K_{0}\left(z\right),$ $\displaystyle\left.\frac{\partial K_{\nu}\left(z\right)}{\partial\nu}\right|_{% \nu=0}$ $\displaystyle=0.$

Half-Integer Values of $\nu$

For the notations $E_{1}$ and $\mathrm{Ei}$ see §6.2(i). When $x>0$,

 10.38.6 $\left.\frac{\partial I_{\nu}\left(x\right)}{\partial\nu}\right|_{\nu=\pm\frac{% 1}{2}}=-\frac{1}{\sqrt{2\pi x}}\left(E_{1}\left(2x\right)e^{x}\pm\mathrm{Ei}% \left(2x\right)e^{-x}\right),$
 10.38.7 $\left.\frac{\partial K_{\nu}\left(x\right)}{\partial\nu}\right|_{\nu=\pm\frac{% 1}{2}}=\pm\sqrt{\frac{\pi}{2x}}E_{1}\left(2x\right)e^{x}.$

For further results see Brychkov and Geddes (2005).