# large rho

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## 1—10 of 21 matching pages

##### 1: 33.11 Asymptotic Expansions for Large $\rho$
###### §33.11 Asymptotic Expansions for Large$\rho$
For large $\rho$, with $\ell$ and $\eta$ fixed, …
##### 3: 2.11 Remainder Terms; Stokes Phenomenon
For large $\rho$ the integrand has a saddle point at $t=e^{-i\theta}$. … 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. …
##### 6: 10.21 Zeros
With $a=(t+\tfrac{1}{2}\nu-\tfrac{1}{4})\pi$, the right-hand side is the asymptotic expansion of $\rho_{\nu}(t)$ for large $t$. …
##### 7: 19.36 Methods of Computation
Descending Gauss transformations of $\Pi\left(\phi,\alpha^{2},k\right)$ (see (19.8.20)) are used in Fettis (1965) to compute a large table (see §19.37(iii)). This method loses significant figures in $\rho$ if $\alpha^{2}$ and $k^{2}$ are nearly equal unless they are given exact values—as they can be for tables. …
##### 8: 2.8 Differential Equations with a Parameter
2.8.9 $\frac{{\mathrm{d}}^{2}W}{{\mathrm{d}\xi}^{2}}=\left(\frac{u^{2}}{\xi}+\frac{% \rho}{\xi^{2}}\right)W,$
##### 9: 10.72 Mathematical Applications
In regions in which the function $f(z)$ has a simple pole at $z=z_{0}$ and $(z-z_{0})^{2}g(z)$ is analytic at $z=z_{0}$ (the case $\lambda=-1$ in §10.72(i)), asymptotic expansions of the solutions $w$ of (10.72.1) for large $u$ can be constructed in terms of Bessel functions and modified Bessel functions of order $\pm\sqrt{1+4\rho}$, where $\rho$ is the limiting value of $(z-z_{0})^{2}g(z)$ as $z\to z_{0}$. …
##### 10: 33.9 Expansions in Series of Bessel Functions
###### §33.9(i) Spherical Bessel Functions
The series (33.9.1) converges for all finite values of $\eta$ and $\rho$.
###### §33.9(ii) Bessel Functions and Modified Bessel Functions
With $t=2\left|\eta\right|\rho$, … Next, as $\eta\to+\infty$ with $\rho$ ($>0$) fixed, …