# large a

(0.004 seconds)

## 1—10 of 155 matching pages

##### 1: Bibliography U
• F. Ursell (1972) Integrals with a large parameter. Several nearly coincident saddle-points. Proc. Cambridge Philos. Soc. 72, pp. 49–65.
• F. Ursell (1980) Integrals with a large parameter: A double complex integral with four nearly coincident saddle-points. Math. Proc. Cambridge Philos. Soc. 87 (2), pp. 249–273.
• F. Ursell (1984) Integrals with a large parameter: Legendre functions of large degree and fixed order. Math. Proc. Cambridge Philos. Soc. 95 (2), pp. 367–380.
• ##### 2: Karl Dilcher
Over the years he authored or coauthored numerous papers on Bernoulli numbers and related topics, and he maintains a large on-line bibliography on the subject. …
##### 3: 15.12 Asymptotic Approximations
If $|\operatorname{ph}\left(z-1\right)|<\pi$, then as $\lambda\to\infty$ with $|\operatorname{ph}\lambda|\leq\pi-\delta$, … If $|\operatorname{ph}z|<\pi$, then as $\lambda\to\infty$ with $|\operatorname{ph}\lambda|\leq\pi-\delta$, … If $|\operatorname{ph}z|<\pi$, then as $\lambda\to\infty$ with $|\operatorname{ph}\lambda|\leq\tfrac{1}{2}\pi-\delta$, … For other extensions, see Wagner (1986), Temme (2003) and Temme (2015, Chapters 12 and 28).
##### 5: 10.70 Zeros
If $m$ is a large positive integer, then …
##### 6: 13.9 Zeros
where $n$ is a large positive integer, and the logarithm takes its principal value (§4.2(i)). … For fixed $b$ and $z$ in $\mathbb{C}$ the large $a$-zeros of $M\left(a,b,z\right)$ are given by …where $n$ is a large positive integer. … For fixed $b$ and $z$ in $\mathbb{C}$ the large $a$-zeros of $U\left(a,b,z\right)$ are given by …where $n$ is a large positive integer. …
##### 7: 8.11 Asymptotic Approximations and Expansions
###### §8.11(iv) Large$a$, Bounded $(x-a)/(2a)^{\frac{1}{2}}$
Large values of $|a|$ or $|b|$, for example, delay convergence of the Gauss series, and may also lead to severe cancellation. … Initial values for moderate values of $|a|$ and $|b|$ can be obtained by the methods of §15.19(i), and for large values of $|a|$, $|b|$, or $|c|$ via the asymptotic expansions of §§15.12(ii) and 15.12(iii). For example, in the half-plane $\Re z\leq\frac{1}{2}$ we can use (15.12.2) or (15.12.3) to compute $F\left(a,b;c+N+1;z\right)$ and $F\left(a,b;c+N;z\right)$, where $N$ is a large positive integer, and then apply (15.5.18) in the backward direction. …
where $z$ is a real or complex variable and $u$ is a large real or complex parameter. … In regions in which (10.72.1) has a simple turning point $z_{0}$, that is, $f(z)$ and $g(z)$ are analytic (or with weaker conditions if $z=x$ is a real variable) and $z_{0}$ is a simple zero of $f(z)$, asymptotic expansions of the solutions $w$ for large $u$ can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order $\tfrac{1}{3}$9.6(i)). … 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 $z\to z_{0}$. …