# change of parameter

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## 11—20 of 54 matching pages

##### 11: 8.11 Asymptotic Approximations and Expansions
8.11.10 $P\left(a+1,x\right)=\tfrac{1}{2}\operatorname{erfc}\left(-y\right)-\frac{1}{3}% \sqrt{\frac{2}{\pi a}}(1+y^{2})e^{-y^{2}}+O\left(a^{-1}\right),$
8.11.11 $\gamma^{*}\left(1-a,-x\right)=x^{a-1}\left(-\cos\left(\pi a\right)+\frac{\sin% \left(\pi a\right)}{\pi}\left(2\sqrt{\pi}F\left(y\right)+\frac{2}{3}\sqrt{% \frac{2\pi}{a}}\left(1-y^{2}\right)\right)e^{y^{2}}+O\left(a^{-1}\right)\right),$
##### 12: 29.2 Differential Equations
29.2.4 $(1-k^{2}{\cos}^{2}\phi)\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}\phi}^{2}}+k^{2}% \cos\phi\sin\phi\frac{\mathrm{d}w}{\mathrm{d}\phi}+(h-\nu(\nu+1)k^{2}{\cos}^{2% }\phi)w=0,$
29.2.5 $\phi=\tfrac{1}{2}\pi-\operatorname{am}\left(z,k\right).$
##### 13: 19.25 Relations to Other Functions
The three changes of parameter of $\Pi\left(\phi,\alpha^{2},k\right)$ in §19.7(iii) are unified in (19.21.12) by way of (19.25.14). …
19.8.14 $2(k^{2}-\alpha^{2})\Pi\left(\phi,\alpha^{2},k\right)=\frac{\omega^{2}-\alpha^{% 2}}{1+k^{\prime}}\Pi\left(\phi_{1},\alpha_{1}^{2},k_{1}\right)+k^{2}F\left(% \phi,k\right)-{(1+k^{\prime})\alpha_{1}^{2}R_{C}\left(c_{1},c_{1}-\alpha_{1}^{% 2}\right)},$
19.8.20 $\rho\Pi\left(\phi,\alpha^{2},k\right)=\frac{4}{1+k^{\prime}}\Pi\left(\psi_{1},% \alpha_{1}^{2},k_{1}\right)+(\rho-1)F\left(\phi,k\right)-R_{C}\left(c-1,c-% \alpha^{2}\right),$
##### 15: 15.8 Transformations of Variable
15.8.32 $\frac{\left(1-z^{3}\right)^{a}}{\left(-z\right)^{3a}}\left(\frac{1}{\Gamma% \left(a+\frac{2}{3}\right)\Gamma\left(\frac{2}{3}\right)}F\left({a,a+\frac{1}{% 3}\atop\frac{2}{3}};z^{-3}\right)+\frac{{\mathrm{e}}^{\frac{1}{3}\pi\mathrm{i}% }}{z\Gamma\left(a\right)\Gamma\left(\frac{4}{3}\right)}F\left({a+\frac{1}{3},a% +\frac{2}{3}\atop\frac{4}{3}};z^{-3}\right)\right)=\frac{3^{\frac{3}{2}a+\frac% {1}{2}}{\mathrm{e}}^{\frac{1}{2}a\pi\mathrm{i}}\Gamma\left(a+\frac{1}{3}\right% )(1-\zeta)^{a}}{2\pi\Gamma\left(2a+\frac{2}{3}\right)(-\zeta)^{2a}}F\left({a+% \frac{1}{3},3a\atop 2a+\frac{2}{3}};\zeta^{-1}\right),$ $|z|>1$, $|\operatorname{ph}\left(-z\right)|<\frac{1}{3}\pi$.
##### 16: 28.2 Definitions and Basic Properties
28.2.2 $\zeta(1-\zeta)w^{\prime\prime}+\tfrac{1}{2}\left(1-2\zeta)w^{\prime}+\tfrac{1}% {4}(a-2q(1-2\zeta)\right)w=0.$
28.2.3 $(1-\zeta^{2})w^{\prime\prime}-\zeta w^{\prime}+\left(a+2q-4q\zeta^{2}\right)w=0.$
##### 17: 20.10 Integrals
###### §20.10(ii) Laplace Transforms with respect to the Lattice Parameter
Then
20.10.4 $\int_{0}^{\infty}e^{-st}\theta_{1}\left(\frac{\beta\pi}{2\ell}\middle|\frac{i% \pi t}{\ell^{2}}\right)\mathrm{d}t=\int_{0}^{\infty}e^{-st}\theta_{2}\left(% \frac{(1+\beta)\pi}{2\ell}\middle|\frac{i\pi t}{\ell^{2}}\right)\mathrm{d}t=-% \frac{\ell}{\sqrt{s}}\sinh\left(\beta\sqrt{s}\right)\operatorname{sech}\left(% \ell\sqrt{s}\right),$
20.10.5 $\int_{0}^{\infty}e^{-st}\theta_{3}\left(\frac{(1+\beta)\pi}{2\ell}\middle|% \frac{i\pi t}{\ell^{2}}\right)\mathrm{d}t=\int_{0}^{\infty}e^{-st}\theta_{4}% \left(\frac{\beta\pi}{2\ell}\middle|\frac{i\pi t}{\ell^{2}}\right)\mathrm{d}t=% \frac{\ell}{\sqrt{s}}\cosh\left(\beta\sqrt{s}\right)\operatorname{csch}\left(% \ell\sqrt{s}\right).$
##### 18: 3.8 Nonlinear Equations
Then the sensitivity of a simple zero $z$ to changes in $\alpha$ is given by …
##### 19: 31.7 Relations to Other Functions
31.7.1 ${{}_{2}F_{1}}\left(\alpha,\beta;\gamma;z\right)=\mathit{H\!\ell}\left(1,\alpha% \beta;\alpha,\beta,\gamma,\delta;z\right)=\mathit{H\!\ell}\left(0,0;\alpha,% \beta,\gamma,\alpha+\beta+1-\gamma;z\right)=\mathit{H\!\ell}\left(a,a\alpha% \beta;\alpha,\beta,\gamma,\alpha+\beta+1-\gamma;z\right).$
Other reductions of $\mathit{H\!\ell}$ to a ${{}_{2}F_{1}}$, with at least one free parameter, exist iff the pair $(a,p)$ takes one of a finite number of values, where $q=\alpha\beta p$. Below are three such reductions with three and two parameters. …
31.7.2 $\mathit{H\!\ell}\left(2,\alpha\beta;\alpha,\beta,\gamma,\alpha+\beta-2\gamma+1% ;z\right)={{}_{2}F_{1}}\left(\tfrac{1}{2}\alpha,\tfrac{1}{2}\beta;\gamma;1-(1-% z)^{2}\right),$
With $z={\operatorname{sn}}^{2}\left(\zeta,k\right)$ and …
##### 20: 28.14 Fourier Series
28.14.5 $\sum_{m=-\infty}^{\infty}\left(c_{2m}^{\nu}(q)\right)^{2}=1;$
For changes of sign of $\nu$, $q$, and $m$,