# solution by variation of parameters

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##### 2: 9.12 Scorer Functions
Solutions of this equation are the Scorer functions and can be found by the method of variation of parameters1.13(iii)). …
##### 3: 10.15 Derivatives with Respect to Order
10.15.2 $\frac{\partial Y_{\nu}\left(z\right)}{\partial\nu}=\cot\left(\nu\pi\right)% \left(\frac{\partial J_{\nu}\left(z\right)}{\partial\nu}-\pi Y_{\nu}\left(z% \right)\right)-\csc\left(\nu\pi\right)\frac{\partial J_{-\nu}\left(z\right)}{% \partial\nu}-\pi J_{\nu}\left(z\right).$
10.15.3 $\left.\frac{\partial J_{\nu}\left(z\right)}{\partial\nu}\right|_{\nu=n}=\frac{% \pi}{2}Y_{n}\left(z\right)+\frac{n!}{2(\tfrac{1}{2}z)^{n}}\sum_{k=0}^{n-1}% \frac{(\tfrac{1}{2}z)^{k}J_{k}\left(z\right)}{k!(n-k)}.$
10.15.4 $\left.\frac{\partial Y_{\nu}\left(z\right)}{\partial\nu}\right|_{\nu=n}=-\frac% {\pi}{2}J_{n}\left(z\right)+\frac{n!}{2(\tfrac{1}{2}z)^{n}}\sum_{k=0}^{n-1}% \frac{(\tfrac{1}{2}z)^{k}Y_{k}\left(z\right)}{k!(n-k)},$
10.15.5 $\left.\frac{\partial J_{\nu}\left(z\right)}{\partial\nu}\right|_{\nu=0}=\frac{% \pi}{2}Y_{0}\left(z\right),\quad\left.\frac{\partial Y_{\nu}\left(z\right)}{% \partial\nu}\right|_{\nu=0}=-\frac{\pi}{2}J_{0}\left(z\right).$
10.15.6 $\left.\frac{\partial J_{\nu}\left(x\right)}{\partial\nu}\right|_{\nu=\frac{1}{% 2}}=\sqrt{\frac{2}{\pi x}}\left(\operatorname{Ci}\left(2x\right)\sin x-% \operatorname{Si}\left(2x\right)\cos x\right),$
##### 4: 2.8 Differential Equations with a Parameter
###### §2.8(i) Classification of Cases
Zeros of $f(z)$ are also called turning points. …
##### 5: Bibliography M
• R. S. Maier (2007) The 192 solutions of the Heun equation. Math. Comp. 76 (258), pp. 811–843.
• M. Mazzocco (2001a) Rational solutions of the Painlevé VI equation. J. Phys. A 34 (11), pp. 2281–2294.
• M. Mazzocco (2001b) Picard and Chazy solutions to the Painlevé VI equation. Math. Ann. 321 (1), pp. 157–195.
• R. J. Muirhead (1978) Latent roots and matrix variates: A review of some asymptotic results. Ann. Statist. 6 (1), pp. 5–33.
• M. E. Muldoon (1981) The variation with respect to order of zeros of Bessel functions. Rend. Sem. Mat. Univ. Politec. Torino 39 (2), pp. 15–25.