# simple

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##### 1: 9.15 Mathematical Applications
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. …
##### 2: 10.72 Mathematical Applications
###### 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 $\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 $z\to z_{0}$. … 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
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). …
##### 4: 27.18 Methods of Computation: Primes
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). …
##### 5: 5.2 Definitions
It is a meromorphic function with no zeros, and with simple poles of residue $(-1)^{n}/n!$ at $z=-n$. $1/\Gamma\left(z\right)$ is entire, with simple zeros at $z=-n$. …$\psi\left(z\right)$ is meromorphic with simple poles of residue $-1$ at $z=-n$. …
##### 6: 31.6 Path-Multiplicative Solutions
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 ${\mathrm{e}}^{2\nu\pi i}$. …
##### 7: Bibliography D
• A. M. Din (1981) A simple sum formula for Clebsch-Gordan coefficients. Lett. Math. Phys. 5 (3), pp. 207–211.
• B. I. Dunlap and B. R. Judd (1975) Novel identities for simple $n$-$j$ symbols. J. Mathematical Phys. 16, pp. 318–319.
• T. M. Dunster (1994b) 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.
• T. M. Dunster (2001a) 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.
• T. M. Dunster (2014) Olver’s error bound methods applied to linear ordinary differential equations having a simple turning point. Anal. Appl. (Singap.) 12 (4), pp. 385–402.
• ##### 8: 3.8 Nonlinear Equations
###### §3.8 Nonlinear Equations
If $f(z_{0})=0$ and $f^{\prime}(z_{0})\neq 0$, 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 $p=\frac{1}{2}(1+\sqrt{5})=1.618\ldots\,$. … Then the sensitivity of a simple zero $z$ to changes in $\alpha$ is given by …
##### 9: 8.27 Approximations
• DiDonato (1978) gives a simple approximation for the function $F(p,x)=x^{-p}e^{x^{2}/2}\int_{x}^{\infty}e^{-t^{2}/2}t^{p}\,\mathrm{d}t$ (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 $x\to\infty$.

• ##### 10: 11.13 Methods of Computation
For simple and effective approximations to $\mathbf{H}_{0}\left(z\right)$ and $\mathbf{H}_{1}\left(z\right)$ see Aarts and Janssen (2016). …