# simple

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## 1—10 of 67 matching pages

##### 1: 9.15 Mathematical Applications

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►Airy functions play an indispensable role in the construction of uniform asymptotic expansions for contour integrals with coalescing saddle points, and for solutions of linear second-order ordinary differential equations with a simple turning point.
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##### 2: 10.72 Mathematical Applications

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###### Simple Turning Points

►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)). … ►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 $z\to {z}_{0}$. … ►In (10.72.1) assume $f(z)=f(z,\alpha )$ and $g(z)=g(z,\alpha )$ depend continuously on a real parameter $\alpha $, $f(z,\alpha )$ has a simple zero $z={z}_{0}(\alpha )$ and a double pole $z=0$, except for a critical value $\alpha =a$, where ${z}_{0}(a)=0$. …##### 3: 13.27 Mathematical Applications

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►For applications of Whittaker functions to the uniform asymptotic theory of differential equations with a coalescing turning point and simple pole see §§2.8(vi) and 18.15(i).
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##### 4: 27.18 Methods of Computation: Primes

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►Two simple algorithms for proving primality require a knowledge of all or part of the factorization of $n-1,n+1$, or both; see Crandall and Pomerance (2005, §§4.1–4.2).
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##### 5: 5.2 Definitions

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►It is a meromorphic function with no zeros, and with simple poles of residue ${(-1)}^{n}/n!$ at $z=-n$.
$1/\mathrm{\Gamma}\left(z\right)$ is entire, with simple zeros at $z=-n$.
…$\psi \left(z\right)$ is meromorphic with simple poles of residue $-1$ at $z=-n$.
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##### 6: 31.6 Path-Multiplicative Solutions

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►This denotes a set of solutions of (31.2.1) with the property that if we pass around a simple closed contour in the $z$-plane that encircles ${s}_{1}$ and ${s}_{2}$ once in the positive sense, but not the remaining finite singularity, then the solution is multiplied by a constant factor ${\mathrm{e}}^{2\nu \pi \mathrm{i}}$.
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##### 7: Bibliography D

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A simple sum formula for Clebsch-Gordan coefficients.
Lett. Math. Phys. 5 (3), pp. 207–211.
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Novel identities for simple
$n$-$j$ symbols.
J. Mathematical Phys. 16, pp. 318–319.
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Uniform asymptotic solutions of second-order linear differential equations having a simple pole and a coalescing turning point in the complex plane.
SIAM J. Math. Anal. 25 (2), pp. 322–353.
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Convergent expansions for solutions of linear ordinary differential equations having a simple turning point, with an application to Bessel functions.
Stud. Appl. Math. 107 (3), pp. 293–323.
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Olver’s error bound methods applied to linear ordinary differential equations having a simple turning point.
Anal. Appl. (Singap.) 12 (4), pp. 385–402.
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##### 8: 3.8 Nonlinear Equations

###### §3.8 Nonlinear Equations

… ►If $f({z}_{0})=0$ and ${f}^{\prime}({z}_{0})\ne 0$, then ${z}_{0}$ is a*simple zero*of $f$. … ►If $\zeta $ is a simple zero, then the iteration converges locally and quadratically. … ►If the wanted zero $\xi $ is simple, then the method converges locally with order of convergence $p=\frac{1}{2}(1+\sqrt{5})=1.618\mathrm{\dots}$. … ►Then the sensitivity of a simple zero $z$ to changes in $\alpha $ is given by …

##### 9: 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 $x\to \mathrm{\infty}$.