for large |γ2|

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31: 3.8 Nonlinear Equations

… ►for all $n$ sufficiently large, where $A$ and $p$ are independent of $n$, then the sequence is said to have convergence of the $p$th order. … ►On the last iteration $q_n{z}^{n-2}+q_{n-1}{z}^{n-3}+\mathrm{\cdots }+q_2$ is the quotient on dividing $p(z)$ by $z^2-sz-t$. … ► $p(z)=z^4-2z^2+1$. With the starting values $s_0=\frac{7}{4}$, $t_0=-\frac{1}{2}$, an approximation to the quadratic factor $z^2-2z+1={(z-1)}^2$ is computed ($s=2$, $t=-1$). … ►For moderate or large values of $n$ it is not uncommon for the magnitude of the right-hand side of (3.8.14) to be very large compared with unity, signifying that the computation of zeros of polynomials is often an ill-posed problem. …

32: 30.11 Radial Spheroidal Wave Functions

… ►with $J_{\nu }$, $Y_{\nu }$, $H_{\nu }^{(1)}$, and $H_{\nu }^{(2)}$ as in §10.2(ii). …Here $a_{n,k}^{-m}({\gamma }^2)$ is defined by (30.8.2) and (30.8.6), and …In (30.11.3) $z\ne 0$ when $j=1$, and $|z|>1$ when $j=2,3,4$. … ►

§30.11(iii) Asymptotic Behavior

►For fixed $\gamma$, as $z\to \mathrm{\infty }$ in the sector $|\mathrm{ph}z|\le \pi -\delta$ (), …

33: 16.11 Asymptotic Expansions

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§16.11(ii) Expansions for Large Variable

… ►The formal series (16.11.2) for $H_{q+1,q}(z)$ converges if $\left|z\right|>1$, and … ►As $z\to \mathrm{\infty }$ in $|\mathrm{ph}z|\le \pi$, … ►As $z\to \mathrm{\infty }$ in $|\mathrm{ph}z|\le \pi$, … ►

§16.11(iii) Expansions for Large Parameters

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34: 32.11 Asymptotic Approximations for Real Variables

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If , then $w(x)$ oscillates about, and is asymptotic to, $-\sqrt{\frac{1}{6}|x|}$ as $x\to -\mathrm{\infty }$.

… ►If , then $w_k(x)$ exists for all sufficiently large $|x|$ as $x\to -\mathrm{\infty }$, and … ►If $|k|=1$, then … ►If $|k|>1$, then $w_k(x)$ has a pole at a finite point $x=c_0$, dependent on $k$, and … ►where $B$ and $\sigma$ are arbitrary constants such that $B\ne 0$ and . …

35: 28.7 Analytic Continuation of Eigenvalues

… ►To 4D the first branch points between $a_0\left(q\right)$ and $a_2\left(q\right)$ are at $q_0=\pm \mathrm{i}1.4688$ with $a_0\left(q_0\right)=a_2\left(q_0\right)=2.0886$, and between $b_2\left(q\right)$ and $b_4\left(q\right)$ they are at $q_1=\pm \mathrm{i}6.9289$ with $b_2\left(q_1\right)=b_4\left(q_1\right)=11.1904$. For real $q$ with , $a_0\left(\mathrm{i}q\right)$ and $a_2\left(\mathrm{i}q\right)$ are real-valued, whereas for real $q$ with $|q|>|q_0|$, $a_0\left(\mathrm{i}q\right)$ and $a_2\left(\mathrm{i}q\right)$ are complex conjugates. … ►For a visualization of the first branch point of $a_0\left(\mathrm{i}\widehat{q}\right)$ and $a_2\left(\mathrm{i}\widehat{q}\right)$ see Figure 28.7.1. … ►All the $a_{2n}\left(q\right)$, $n=0,1,2,\mathrm{\dots }$, can be regarded as belonging to a complete analytic function (in the large). …Analogous statements hold for $a_{2n+1}\left(q\right)$, $b_{2n+1}\left(q\right)$, and $b_{2n+2}\left(q\right)$, also for $n=0,1,2,\mathrm{\dots }$. …

36: 8.13 Zeros

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one negative zero $x_{-}(a)$ and no positive zeros when ;

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one negative zero $x_{-}(a)$ and one positive zero $x_{+}(a)$ when .

… ►For information on the distribution and computation of zeros of $\gamma (a,\lambda a)$ and $\mathrm{\Gamma }(a,\lambda a)$ in the complex $\lambda$-plane for large values of the positive real parameter $a$ see Temme (1995a). … ►

two zeros in each of the intervals when ;

… ►Approximations to $a_n^{\ast }$, $x_n^{\ast }$ for large $n$ can be found in Kölbig (1970). …

37: 2.10 Sums and Sequences

… ►for large $n$. … ►As a first estimate for large $n$ … ►Hence …the last step following from $|x^t|\le 1$ when $t$ is on the interval $[-\frac{1}{2},0]$, the imaginary axis, or the small semicircle. … ►

Example

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38: 8.27 Approximations

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DiDonato (1978) gives a simple approximation for the function $F(p,x)=x^{-p}{\mathrm{e}}^{x^2/2}{\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{-t^2/2}{t}^{p}dt$ (which is related to the incomplete gamma function by a change of variables) for real $p$ and large positive $x$. This takes the form $F(p,x)=4x/h(p,x)$, approximately, where $h(p,x)=3(x^2-p)+\sqrt{{(x^2-p)}^2+8(x^2+p)}$ and is shown to produce an absolute error $O\left(x^{-7}\right)$ as $x\to \mathrm{\infty }$.

… ►

Luke (1969b, p. 186) gives hypergeometric polynomial representations that converge uniformly on compact subsets of the $z$-plane that exclude $z=0$ and are valid for .

… ►

Luke (1975, p. 106) gives rational and Padé approximations, with remainders, for $E_1\left(z\right)$ and $z^{-1}{\int }_0^z{t}^{-1}(1-{\mathrm{e}}^{-t})dt$ for complex $z$ with $\left|\mathrm{ph}z\right|\le \pi$.

…

39: 13.21 Uniform Asymptotic Approximations for Large $\kappa$

§13.21 Uniform Asymptotic Approximations for Large $\kappa$

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§13.21(i) Large $\kappa$, Fixed $\mu$

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§13.21(iv) Large $\kappa$, Other Expansions

►For a uniform asymptotic expansion in terms of Airy functions for $W_{\kappa ,\mu }\left(4\kappa x\right)$ when $\kappa$ is large and positive, $\mu$ is real with $|\mu |$ bounded, and $x\in [\delta ,\mathrm{\infty })$ see Olver (1997b, Chapter 11, Ex. 7.3). … ►

40: 13.29 Methods of Computation

… ►Although the Maclaurin series expansion (13.2.2) converges for all finite values of $z$, it is cumbersome to use when $|z|$ is large owing to slowness of convergence and cancellation. For large $|z|$ the asymptotic expansions of §13.7 should be used instead. …For large values of the parameters $a$ and $b$ the approximations in §13.8 are available. … ►For $U(a,b,z)$ and $W_{\kappa ,\mu }\left(z\right)$ we may integrate along outward rays from the origin in the sectors , with initial values obtained from connection formulas in §13.2(vii), §13.14(vii). In the sector the integration has to be towards the origin, with starting values computed from asymptotic expansions (§§13.7 and 13.19). …