# large a

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## 1—10 of 148 matching pages

##### 1: Bibliography U

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Integrals with a large parameter. Several nearly coincident saddle-points.
Proc. Cambridge Philos. Soc. 72, pp. 49–65.
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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.
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Integrals with a large parameter: Legendre functions of large degree and fixed order.
Math. Proc. Cambridge Philos. Soc. 95 (2), pp. 367–380.
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##### 2: Karl Dilcher

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►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.
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##### 3: 15.12 Asymptotic Approximations

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►If $$, then as $\lambda \to \mathrm{\infty}$ with $|\mathrm{ph}\lambda |\le \pi -\delta $,
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►If $$, then as $\lambda \to \mathrm{\infty}$ with $|\mathrm{ph}\lambda |\le \pi -\delta $,
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►If $$, then as $\lambda \to \mathrm{\infty}$ with $|\mathrm{ph}\lambda |\le \frac{1}{2}\pi -\delta $,
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►For other extensions, see Wagner (1986), Temme (2003) and Temme (2015, Chapters 12 and 28).

##### 4: 10.70 Zeros

##### 5: 13.9 Zeros

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►where $n$ is a large positive integer, and the logarithm takes its principal value (§4.2(i)).
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►For fixed $b$ and $z$ in $\u2102$ the large
$a$-zeros of $M(a,b,z)$ are given by
…where $n$ is a large positive integer.
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►For fixed $b$ and $z$ in $\u2102$ the large
$a$-zeros of $U(a,b,z)$ are given by
…where $n$ is a large positive integer.
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##### 6: 8.11 Asymptotic Approximations and Expansions

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###### §8.11(i) Large $z$, Fixed $a$

… ►###### §8.11(ii) Large $a$, Fixed $z$

… ►###### §8.11(iii) Large $a$, Fixed $z/a$

… ►###### §8.11(iv) Large $a$, Bounded $(x-a)/{(2a)}^{\frac{1}{2}}$

…##### 7: About the Project

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►These products resulted from the leadership of the Editors and Associate Editors pictured in Figure 1; the contributions of 29 authors, 10 validators, and 5 principal developers; and assistance from a large group of contributing developers, consultants, assistants and interns.
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##### 8: 13.8 Asymptotic Approximations for Large Parameters

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###### §13.8(i) Large $|b|$, Fixed $a$ and $z$

… ►When the foregoing results are combined with Kummer’s transformation (13.2.39), an approximation is obtained for the case when $|b|$ is large, and $|b-a|$ and $|z|$ are bounded. ►###### §13.8(ii) Large $b$ and $z$, Fixed $a$ and $b/z$

… ►###### §13.8(iii) Large $a$

… ► …##### 9: 15.19 Methods of Computation

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►Large values of $|a|$ or $|b|$, for example, delay convergence of the Gauss series, and may also lead to severe cancellation.
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►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 $\mathrm{\Re}z\le \frac{1}{2}$ we can use (15.12.2) or (15.12.3) to compute $F(a,b;c+N+1;z)$ and $F(a,b;c+N;z)$, where $N$ is a large positive integer, and then apply (15.5.18) in the backward direction.
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##### 10: 10.72 Mathematical Applications

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►where $z$ is a real or complex variable and $u$ is a large real or complex parameter.
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►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 $\frac{1}{3}$ (§9.6(i)).
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►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 $z\to {z}_{0}$.
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