# small rho

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## 1—10 of 16 matching pages

##### 2: 18.15 Asymptotic Approximations
18.15.7 $\varepsilon_{M}(\rho,\theta)=\begin{cases}\theta O\left(\rho^{-2M-(3/2)}\right% ),&c\rho^{-1}\leq\theta\leq\pi-\delta,\\ \theta^{\alpha+(5/2)}O\left(\rho^{-2M+\alpha}\right),&0\leq\theta\leq c\rho^{-% 1},\end{cases}$
##### 3: 2.11 Remainder Terms; Stokes Phenomenon
Owing to the factor $e^{-\rho}$, that is, $e^{-|z|}$ in (2.11.13), $F_{n+p}\left(z\right)$ is uniformly exponentially small compared with $E_{p}\left(z\right)$. … Hence from §7.12(i) $\operatorname{erfc}\left(\sqrt{\frac{1}{2}\rho}\;c(\theta)\right)$ is of the same exponentially-small order of magnitude as the contribution from the other terms in (2.11.15) when $\rho$ is large. On the other hand, when $\pi+\delta\leq\theta\leq 3\pi-\delta$, $c(\theta)$ is in the left half-plane and $\operatorname{erfc}\left(\sqrt{\frac{1}{2}\rho}\;c(\theta)\right)$ differs from 2 by an exponentially-small quantity. …
##### 4: 33.12 Asymptotic Expansions for Large $\eta$
###### §33.12(ii) Uniform Expansions
With the substitution $\rho=2\eta z$, Equation (33.2.1) becomes …
##### 5: 3.2 Linear Algebra
$\|\mathbf{A}\|_{2}=\sqrt{\rho(\mathbf{A}\mathbf{A}^{\rm T})},$
where $\rho(\mathbf{A}\mathbf{A}^{\rm T})$ is the largest of the absolute values of the eigenvalues of the matrix $\mathbf{A}\mathbf{A}^{\rm T}$; see §3.2(iv). … The sensitivity of the solution vector $\mathbf{x}$ in (3.2.1) to small perturbations in the matrix $\mathbf{A}$ and the vector $\mathbf{b}$ is measured by the condition numberIf $\mathbf{A}$ is nondefective and $\lambda$ is a simple zero of $p_{n}(\lambda)$, then the sensitivity of $\lambda$ to small perturbations in the matrix $\mathbf{A}$ is measured by the condition number
##### 6: 33.23 Methods of Computation
###### §33.23 Methods of Computation
Thus the regular solutions can be computed from the power-series expansions (§§33.6, 33.19) for small values of the radii and then integrated in the direction of increasing values of the radii. …
###### §33.23(vii) WKBJ Approximations
WKBJ approximations (§2.7(iii)) for $\rho>\rho_{\operatorname{tp}}\left(\eta,\ell\right)$ are presented in Hull and Breit (1959) and Seaton and Peach (1962: in Eq.  (12) $(\rho-c)/c$ should be $(\rho-c)/\rho$). …
##### 7: 27.19 Methods of Computation: Factorization
Type I probabilistic algorithms include the Brent–Pollard rho algorithm (also called Monte Carlo method), the Pollard $p-1$ algorithm, and the Elliptic Curve Method (ecm). …As of January 2009 the largest prime factors found by these methods are a 19-digit prime for Brent–Pollard rho, a 58-digit prime for Pollard $p-1$, and a 67-digit prime for ecm. … The snfs can be applied only to numbers that are very close to a power of a very small base. …
##### 8: 2.5 Mellin Transform Methods
###### §2.5(iii) Laplace Transforms with Small Parameters
for any $\rho$ satisfying $1<\rho<2$. … where $l$ ($\geq 2$) is an arbitrary integer and $\delta$ is an arbitrary small positive constant. … … To verify (2.5.48) we may use …
##### 9: 28.35 Tables
• Blanch and Clemm (1969) includes eigenvalues $a_{n}\left(q\right)$, $b_{n}\left(q\right)$ for $q=\rho e^{\mathrm{i}\phi}$, $\rho=0(.5)25$, $\phi=5^{\circ}(5^{\circ})90^{\circ}$, $n=0(1)15$; 4D. Also $a_{n}\left(q\right)$ and $b_{n}\left(q\right)$ for $q=\mathrm{i}\rho$, $\rho=0(.5)100$, $n=0(2)14$ and $n=2(2)16$, respectively; 8D. Double points for $n=0(1)15$; 8D. Graphs are included.

• Ince (1932) includes the first zero for $\operatorname{ce}_{n}$, $\operatorname{se}_{n}$ for $n=2(1)5$ or $6$, $q=0(1)10(2)40$; 4D. This reference also gives zeros of the first derivatives, together with expansions for small $q$.

• ##### 10: 13.7 Asymptotic Expansions for Large Argument
Here $\delta$ denotes an arbitrary small positive constant. …
13.7.3 $U\left(a,b,z\right)\sim z^{-a}\sum_{s=0}^{\infty}\frac{{\left(a\right)_{s}}{% \left(a-b+1\right)_{s}}}{s!}(-z)^{-s},$ $|\operatorname{ph}z|\leq\tfrac{3}{2}\pi-\delta$.
13.7.5 $\left|\varepsilon_{n}(z)\right|,~{}\beta^{-1}\left|\varepsilon_{n}^{\prime}(z)% \right|\leq 2\alpha C_{n}\left|\frac{{\left(a\right)_{n}}{\left(a-b+1\right)_{% n}}}{n!z^{a+n}}\right|\exp\left(\frac{2\alpha\rho C_{1}}{|z|}\right),$
$\rho=\tfrac{1}{2}\left|2a^{2}-2ab+b\right|+\frac{\sigma(1+\frac{1}{4}\sigma)}{% (1-\sigma)^{2}}$ ,
13.7.13 $R_{m,n}(a,b,z)=\begin{cases}O\left(e^{-|z|}z^{-m}\right),&|\operatorname{ph}z|% \leq\pi,\\ O\left(e^{z}z^{-m}\right),&\pi\leq|\operatorname{ph}z|\leq\tfrac{5}{2}\pi-% \delta.\\ \end{cases}$