# asymptotic expansion

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## 1—10 of 164 matching pages

##### 1: 16.22 Asymptotic Expansions
###### §16.22 AsymptoticExpansions
Asymptotic expansions of ${G^{m,n}_{p,q}}\left(z;\mathbf{a};\mathbf{b}\right)$ for large $z$ are given in Luke (1969a, §§5.7 and 5.10) and Luke (1975, §5.9). For asymptotic expansions of Meijer $G$-functions with large parameters see Fields (1973, 1983).
##### 2: 10.70 Zeros
$\mbox{zeros of \operatorname{ber}_{\nu}x}\sim\sqrt{2}(t-f(t)),$ $t=(m-\tfrac{1}{2}\nu-\tfrac{3}{8})\pi$,
$\mbox{zeros of \operatorname{bei}_{\nu}x}\sim\sqrt{2}(t-f(t)),$ $t=(m-\tfrac{1}{2}\nu+\tfrac{1}{8})\pi$,
$\mbox{zeros of \operatorname{ker}_{\nu}x}\sim\sqrt{2}(t+f(-t)),$ $t=(m-\tfrac{1}{2}\nu-\tfrac{5}{8})\pi$,
$\mbox{zeros of \operatorname{kei}_{\nu}x}\sim\sqrt{2}(t+f(-t)),$ $t=(m-\tfrac{1}{2}\nu-\tfrac{1}{8})\pi$.
##### 3: 10.67 Asymptotic Expansions for Large Argument
###### §10.67 AsymptoticExpansions for Large Argument
10.67.1 $\operatorname{ker}_{\nu}x\sim e^{-x/\sqrt{2}}\left(\frac{\pi}{2x}\right)^{% \frac{1}{2}}\*\sum_{k=0}^{\infty}\frac{a_{k}(\nu)}{x^{k}}\cos\left(\frac{x}{% \sqrt{2}}+\left(\frac{\nu}{2}+\frac{k}{4}+\frac{1}{8}\right)\pi\right),$
###### §10.67(ii) Cross-Products and Sums of Squares in the Case $\nu=0$
10.67.9 ${\operatorname{ber}^{2}}x+{\operatorname{bei}^{2}}x\sim\frac{e^{x\sqrt{2}}}{2% \pi x}\left(1+\frac{1}{4\sqrt{2}}\frac{1}{x}+\frac{1}{64}\frac{1}{x^{2}}-\frac% {33}{256\sqrt{2}}\frac{1}{x^{3}}-\frac{1797}{8192}\frac{1}{x^{4}}+\dotsb\right),$
##### 4: 28.16 Asymptotic Expansions for Large $q$
###### §28.16 AsymptoticExpansions for Large $q$
28.16.1 $\lambda_{\nu}\left(h^{2}\right)\sim-2h^{2}+2sh-\dfrac{1}{8}(s^{2}+1)-\dfrac{1}% {2^{7}h}(s^{3}+3s)-\dfrac{1}{2^{12}h^{2}}(5s^{4}+34s^{2}+9)-\dfrac{1}{2^{17}h^% {3}}(33s^{5}+410s^{3}+405s)-\dfrac{1}{2^{20}h^{4}}(63s^{6}+1260s^{4}+2943s^{2}% +486)-\dfrac{1}{2^{25}h^{5}}(527s^{7}+15617s^{5}+69001s^{3}+41607s)+\cdots.$
##### 5: 10.69 Uniform Asymptotic Expansions for Large Order
###### §10.69 Uniform AsymptoticExpansions for Large Order
10.69.2 $\operatorname{ber}_{\nu}\left(\nu x\right)+i\operatorname{bei}_{\nu}\left(\nu x% \right)\sim\frac{e^{\nu\xi}}{(2\pi\nu\xi)^{\ifrac{1}{2}}}\left(\frac{xe^{3\pi i% /4}}{1+\xi}\right)^{\nu}\sum_{k=0}^{\infty}\frac{U_{k}(\xi^{-1})}{\nu^{k}},$
All fractional powers take their principal values. …
##### 6: 12.9 Asymptotic Expansions for Large Variable
###### §12.9 AsymptoticExpansions for Large Variable
12.9.1 $U\left(a,z\right)\sim e^{-\frac{1}{4}z^{2}}z^{-a-\frac{1}{2}}\sum_{s=0}^{% \infty}(-1)^{s}\frac{{\left(\frac{1}{2}+a\right)_{2s}}}{s!(2z^{2})^{s}},$ $|\operatorname{ph}z|\leq\tfrac{3}{4}\pi-\delta(<\tfrac{3}{4}\pi)$ ,
12.9.2 $V\left(a,z\right)\sim\sqrt{\frac{2}{\pi}}e^{\frac{1}{4}z^{2}}z^{a-\frac{1}{2}}% \sum_{s=0}^{\infty}\frac{{\left(\frac{1}{2}-a\right)_{2s}}}{s!(2z^{2})^{s}},$ $|\operatorname{ph}z|\leq\tfrac{1}{4}\pi-\delta(<\tfrac{1}{4}\pi)$ .
12.9.4 $V\left(a,z\right)\sim\sqrt{\frac{2}{\pi}}e^{\frac{1}{4}z^{2}}z^{a-\frac{1}{2}}% \sum_{s=0}^{\infty}\frac{{\left(\tfrac{1}{2}-a\right)_{2s}}}{s!(2z^{2})^{s}}% \pm\frac{i}{\Gamma\left(\tfrac{1}{2}-a\right)}e^{-\frac{1}{4}z^{2}}z^{-a-\frac% {1}{2}}\sum_{s=0}^{\infty}(-1)^{s}\frac{{\left(\tfrac{1}{2}+a\right)_{2s}}}{s!% (2z^{2})^{s}},$ $-\tfrac{1}{4}\pi+\delta\leq\pm\operatorname{ph}z\leq\tfrac{3}{4}\pi-\delta$.
##### 7: 6.12 Asymptotic Expansions
###### §6.12(i) Exponential and Logarithmic Integrals
For the function $\chi$ see §9.7(i). …
##### 9: 13.19 Asymptotic Expansions for Large Argument
###### §13.19 AsymptoticExpansions for Large Argument
13.19.1 $M_{\kappa,\mu}\left(x\right)\sim\frac{\Gamma\left(1+2\mu\right)}{\Gamma\left(% \frac{1}{2}+\mu-\kappa\right)}e^{\frac{1}{2}x}x^{-\kappa}\*\sum_{s=0}^{\infty}% \frac{{\left(\frac{1}{2}-\mu+\kappa\right)_{s}}{\left(\frac{1}{2}+\mu+\kappa% \right)_{s}}}{s!}x^{-s},$ $\mu-\kappa\neq-\tfrac{1}{2},-\tfrac{3}{2},\dots$.
13.19.3 $W_{\kappa,\mu}\left(z\right)\sim e^{-\frac{1}{2}z}z^{\kappa}\sum_{s=0}^{\infty% }\frac{{\left(\frac{1}{2}+\mu-\kappa\right)_{s}}{\left(\frac{1}{2}-\mu-\kappa% \right)_{s}}}{s!}{(-z)^{-s}},$ $|\operatorname{ph}z|\leq\tfrac{3}{2}\pi-\delta$.
For an asymptotic expansion of $W_{\kappa,\mu}\left(z\right)$ as $z\to\infty$ that is valid in the sector $|\operatorname{ph}z|\leq\pi-\delta$ and where the real parameters $\kappa$, $\mu$ are subject to the growth conditions $\kappa=o\left(z\right)$, $\mu=o\left(\sqrt{z}\right)$, see Wong (1973a).
##### 10: 2.2 Transcendental Equations
2.2.6 $t=y^{\frac{1}{2}}\left(1+\tfrac{1}{4}y^{-1}\ln y+o\left(y^{-1}\right)\right),$ $y\to\infty$.
An important case is the reversion of asymptotic expansions for zeros of special functions. …
2.2.7 $f(x)\sim x+f_{0}+f_{1}x^{-1}+f_{2}x^{-2}+\cdots,$ $x\to\infty$.
2.2.8 $x\sim y-F_{0}-F_{1}y^{-1}-F_{2}y^{-2}-\cdots,$ $y\to\infty$,
where $F_{0}=f_{0}$ and $sF_{s}$ ($s\geq 1$) is the coefficient of $x^{-1}$ in the asymptotic expansion of $(f(x))^{s}$ (Lagrange’s formula for the reversion of series). …