# variation of parameters

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## 1—10 of 11 matching pages

##### 2: 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 $\left.\frac{\partial J_{\nu}\left(x\right)}{\partial\nu}\right|_{\nu=\frac{1}{% 2}}=\sqrt{\frac{2}{\pi x}}\left(\operatorname{Ci}\left(2x\right)\sin x-% \operatorname{Si}\left(2x\right)\cos x\right),$
##### 3: 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)). …
##### 4: 10.40 Asymptotic Expansions for Large Argument
10.40.11 $|R_{\ell}(\nu,z)|\leq 2|a_{\ell}(\nu)|\mathcal{V}_{z,\infty}\left(t^{-\ell}% \right)\*\exp\left(|\nu^{2}-\tfrac{1}{4}|\mathcal{V}_{z,\infty}\left(t^{-1}% \right)\right),$
##### 5: 10.17 Asymptotic Expansions for Large Argument
10.17.14 $\left|R_{\ell}^{\pm}(\nu,z)\right|\leq 2|a_{\ell}(\nu)|\mathcal{V}_{z,\pm i% \infty}\left(t^{-\ell}\right)\*\exp\left(|\nu^{2}-\tfrac{1}{4}|\mathcal{V}_{z,% \pm i\infty}\left(t^{-1}\right)\right),$
##### 6: 2.3 Integrals of a Real Variable
Then … In both cases the $n$th error term is bounded in absolute value by $x^{-n}\mathcal{V}_{a,b}\left(q^{(n-1)}(t)\right)$, where the variational operator $\mathcal{V}_{a,b}$ is defined by … Then the series obtained by substituting (2.3.7) into (2.3.1) and integrating formally term by term yields an asymptotic expansion: … (In other words, differentiation of (2.3.8) with respect to the parameter $\lambda$ (or $\mu$) is legitimate.) … When the parameter $x$ is large the contributions from the real and imaginary parts of the integrand in …
##### 7: Bibliography M
• E. L. Mansfield and H. N. Webster (1998) On one-parameter families of Painlevé III. Stud. Appl. Math. 101 (3), pp. 321–341.
• 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.
• H. P. Mulholland and S. Goldstein (1929) The characteristic numbers of the Mathieu equation with purely imaginary parameter. Phil. Mag. Series 7 8 (53), pp. 834–840.
• J. Murzewski and A. Sowa (1972) Tables of the functions of the parabolic cylinder for negative integer parameters. Zastos. Mat. 13, pp. 261–273.
• ##### 8: 10.23 Sums
10.23.1 $\mathscr{C}_{\nu}\left(\lambda z\right)=\lambda^{\pm\nu}\sum_{k=0}^{\infty}% \frac{(\mp 1)^{k}(\lambda^{2}-1)^{k}(\tfrac{1}{2}z)^{k}}{k!}\mathscr{C}_{\nu% \pm k}\left(z\right),$ $|\lambda^{2}-1|<1$.
10.23.2 $\mathscr{C}_{\nu}\left(u\pm v\right)=\sum_{k=-\infty}^{\infty}\mathscr{C}_{\nu% \mp k}\left(u\right)J_{k}\left(v\right),$ $|v|<|u|$.
10.23.20 $\tfrac{1}{2}f(x-)+\tfrac{1}{2}f(x+)=\sum_{m=1}^{\infty}a_{m}J_{\nu}\left(j_{% \nu,m}x\right),$
provided that $f(t)$ is of bounded variation1.4(v)) on an interval $[a,b]$ with $0. …
##### 9: 1.14 Integral Transforms
Suppose that $f(t)$ is absolutely integrable on $(-\infty,\infty)$ and of bounded variation in a neighborhood of $t=u$1.4(v)). … If $f(t)$ is absolutely integrable on $[0,\infty)$ and of bounded variation1.4(v)) in a neighborhood of $t=u$, then … If $\int^{\infty}_{0}|f(t)|\,\mathrm{d}t<\infty$, $g(t)$ is of bounded variation on $(0,\infty)$ and $g(t)\to 0$ as $t\to\infty$, then … Suppose $f(t)$ is a real- or complex-valued function and $s$ is a real or complex parameter. … Suppose the integral (1.14.32) is absolutely convergent on the line $\Re s=\sigma$ and $f(x)$ is of bounded variation in a neighborhood of $x=u$. …
##### 10: 2.8 Differential Equations with a Parameter
###### §2.8 Differential Equations with a Parameter
In addition, $\mathcal{V}_{\mathscr{Q}_{j}}\left(A_{1}\right)$ and $\mathcal{V}_{\mathscr{Q}_{j}}\left(A_{n}\right)$ must be bounded on $\mathbf{\Delta}_{j}(\alpha_{j})$. … These results are valid when $\mathcal{V}_{\alpha_{1},\alpha_{2}}\left(|\xi|^{1/2}B_{0}\right)$ and $\mathcal{V}_{\alpha_{1},\alpha_{2}}\left(|\xi|^{1/2}B_{n-1}\right)$ are finite. … These results are valid when $\mathcal{V}_{0,\alpha_{2}}\left(\xi^{1/2}B_{0}\right)$ and $\mathcal{V}_{0,\alpha_{2}}\left(\xi^{1/2}B_{n-1}\right)$ are finite. … These results are valid when $\mathcal{V}_{\alpha_{1},0}\left(|\xi|^{1/2}B_{0}\right)$ and $\mathcal{V}_{\alpha_{1},0}\left(|\xi|^{1/2}B_{n-1}\right)$ are finite. …