# small eta

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## 1—10 of 12 matching pages

##### 2: 10.41 Asymptotic Expansions for Large Order
10.41.12 $I_{\nu}\left(\nu z\right)=\frac{e^{\nu\eta}}{(2\pi\nu)^{\frac{1}{2}}(1+z^{2})^% {\frac{1}{4}}}\left(\sum_{k=0}^{\ell-1}\frac{U_{k}(p)}{\nu^{k}}+O\left(\frac{1% }{z^{\ell}}\right)\right),$ $|\operatorname{ph}z|\leq\tfrac{1}{2}\pi-\delta$,
10.41.13 $K_{\nu}\left(\nu z\right)=\left(\frac{\pi}{2\nu}\right)^{\frac{1}{2}}\frac{e^{% -\nu\eta}}{(1+z^{2})^{\frac{1}{4}}}\*\left(\sum_{k=0}^{\ell-1}(-1)^{k}\frac{U_% {k}(p)}{\nu^{k}}+O\left(\frac{1}{z^{\ell}}\right)\right),$ $|\operatorname{ph}z|\leq\tfrac{3}{2}\pi-\delta$.
##### 3: 8.12 Uniform Asymptotic Expansions for Large Parameter
8.12.15 $Q\left(a,a\right)\sim\frac{1}{2}+\frac{1}{\sqrt{2\pi a}}\sum_{k=0}^{\infty}c_{% k}(0)a^{-k},$ $|\operatorname{ph}a|\leq\pi-\delta$,
8.12.16 $\frac{e^{\pm\pi ia}}{2i\sin\left(\pi a\right)}Q\left(-a,ae^{\pm\pi i}\right)% \sim\pm\frac{1}{2}-\frac{i}{\sqrt{2\pi a}}\sum_{k=0}^{\infty}c_{k}(0)(-a)^{-k},$ $|\operatorname{ph}a|\leq\pi-\delta$,
##### 4: 33.12 Asymptotic Expansions for Large $\eta$
###### §33.12(i) Transition Region
Then as $\eta\to\infty$, …
##### 5: 8.18 Asymptotic Expansions of $I_{x}\left(a,b\right)$
uniformly for $x\in(0,1)$ and $a/(a+b)$, $b/(a+b)\in[\delta,1-\delta]$, where $\delta$ again denotes an arbitrary small positive constant. …with $\eta/(x-x_{0})>0$, and
8.18.11 $c_{0}(\eta)=\frac{1}{\eta}-\frac{\sqrt{x_{0}(1-x_{0})}}{x-x_{0}},$
For this result, and for higher coefficients $c_{k}(\eta)$ see Temme (1996b, §11.3.3.2). All of the $c_{k}(\eta)$ are analytic at $\eta=0$. …
##### 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. … Bardin et al. (1972) describes ten different methods for the calculation of $F_{\ell}$ and $G_{\ell}$, valid in different regions of the ($\eta,\rho$)-plane. … Noble (2004) obtains double-precision accuracy for $W_{-\eta,\mu}\left(2\rho\right)$ for a wide range of parameters using a combination of recurrence techniques, power-series expansions, and numerical quadrature; compare (33.2.7).
##### 7: 10.75 Tables
• British Association for the Advancement of Science (1937) tabulates $J_{0}\left(x\right)$, $J_{1}\left(x\right)$, $x=0(.001)16(.01)25$, 10D; $Y_{0}\left(x\right)$, $Y_{1}\left(x\right)$, $x=0.01(.01)25$, 8–9S or 8D. Also included are auxiliary functions to facilitate interpolation of the tables of $Y_{0}\left(x\right)$, $Y_{1}\left(x\right)$ for small values of $x$, as well as auxiliary functions to compute all four functions for large values of $x$.

• British Association for the Advancement of Science (1937) tabulates $I_{0}\left(x\right)$, $I_{1}\left(x\right)$, $x=0(.001)5$, 7–8D; $K_{0}\left(x\right)$, $K_{1}\left(x\right)$, $x=0.01(.01)5$, 7–10D; $e^{-x}I_{0}\left(x\right)$, $e^{-x}I_{1}\left(x\right)$, $e^{x}K_{0}\left(x\right)$, $e^{x}K_{1}\left(x\right)$, $x=5(.01)10(.1)20$, 8D. Also included are auxiliary functions to facilitate interpolation of the tables of $K_{0}\left(x\right)$, $K_{1}\left(x\right)$ for small values of $x$.

• Olver (1962) provides tables for the uniform asymptotic expansions given in §10.41(ii), including $\eta$ and the coefficients $U_{k}(p)$, $V_{k}(p)$ as functions of $p=(1+x^{2})^{-\frac{1}{2}}$. These enable $I_{\nu}\left(\nu x\right)$, $K_{\nu}\left(\nu x\right)$, $I_{\nu}'\left(\nu x\right)$, $K_{\nu}'\left(\nu x\right)$ to be computed to 10S when $\nu\geq 16$.

• Young and Kirk (1964) tabulates $\operatorname{ber}_{n}x$, $\operatorname{bei}_{n}x$, $\operatorname{ker}_{n}x$, $\operatorname{kei}_{n}x$, $n=0,1$, $x=0(.1)10$, 15D; $\operatorname{ber}_{n}x$, $\operatorname{bei}_{n}x$, $\operatorname{ker}_{n}x$, $\operatorname{kei}_{n}x$, modulus and phase functions $M_{n}\left(x\right)$, $\theta_{n}\left(x\right)$, $N_{n}\left(x\right)$, $\phi_{n}\left(x\right)$, $n=0,1,2$, $x=0(.01)2.5$, 8S, and $n=0(1)10$, $x=0(.1)10$, 7S. Also included are auxiliary functions to facilitate interpolation of the tables for $n=0(1)10$ for small values of $x$. (Concerning the phase functions see §10.68(iv).)

• ##### 8: 23.22 Methods of Computation
The modular functions $\lambda\left(\tau\right)$, $J\left(\tau\right)$, and $\eta\left(\tau\right)$ are also obtainable in a similar manner from their definitions in §23.15(ii). … For $2\omega_{3}$ choose a nonzero point that is not a multiple of $2\omega_{1}$ and is such that $\Im\tau>0$ and $|\tau|$ is as small as possible, where $\tau=\omega_{3}/\omega_{1}$. …
##### 9: 28.33 Physical Applications
with $W(x,y,t)=e^{\mathrm{i}\omega t}V(x,y)$, reduces to (28.32.2) with $k^{2}=\omega^{2}\rho/{\tau}$. …The separated solutions $V_{n}(\xi,\eta)$ must be $2\pi$-periodic in $\eta$, and have the form
28.33.2 $V_{n}(\xi,\eta)=\left(c_{n}{\operatorname{M}^{(1)}_{n}}\left(\xi,\sqrt{q}% \right)+d_{n}{\operatorname{M}^{(2)}_{n}}\left(\xi,\sqrt{q}\right)\right)% \operatorname{me}_{n}\left(\eta,q\right),$
However, in response to a small perturbation at least one solution may become unbounded. …
##### 10: 2.4 Contour Integrals
2.4.3 $q(t)=\frac{1}{2\pi i}\lim\limits_{\eta\to\infty}\int_{\sigma-i\eta}^{\sigma+i% \eta}e^{tz}Q(z)\,\mathrm{d}z,$ $0,
2.4.6 $f(t)=\frac{1}{2\pi i}\lim\limits_{\eta\to\infty}\int_{\sigma-i\eta}^{\sigma+i% \eta}e^{tz}F(z)\,\mathrm{d}z$
2.4.7 $q(t)-f(t)=\frac{e^{\sigma t}}{2\pi}\lim\limits_{\eta\to\infty}\int_{-\eta}^{% \eta}e^{it\tau}(Q(\sigma+i\tau)-F(\sigma+i\tau))\,\mathrm{d}\tau.$