simple zero

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1: 10.72 Mathematical Applications

… ►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 (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 $z_0(a)=0$. …

2: 5.2 Definitions

… ►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 $z=-n$. …

3: 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 …

4: 7.13 Zeros

… ► $\mathrm{erf}z$ has a simple zero at $z=0$, and in the first quadrant of $ℂ$ there is an infinite set of zeros $z_n=x_n+\mathrm{i}{y}_n$, $n=1,2,3,\mathrm{\dots }$, arranged in order of increasing absolute value. … ►At $z=0$, $C\left(z\right)$ has a simple zero and $S\left(z\right)$ has a triple zero. …

5: 1.10 Functions of a Complex Variable

… ►When $m=1$ the zero is simple. … ►If $f(z)$ and $g(z)$ are analytic on and inside a simple closed contour $C$, and on $C$, then $f(z)$ and $f(z)+g(z)$ have the same number of zeros inside $C$. …

6: 2.4 Contour Integrals

… ►In the commonest case the interior minimum $t_0$ of $\mathrm{\Re }\left(zp(t)\right)$ is a simple zero of $p^{\prime }(t)$. … ►Suppose that on the integration path $𝒫$ there are two simple zeros of $\partial p(\alpha ,t)/\partial t$ that coincide for a certain value $\widehat{\alpha }$ of $\alpha$. …

7: 22.2 Definitions

… ►Each is meromorphic in $z$ for fixed $k$, with simple poles and simple zeros, and each is meromorphic in $k$ for fixed $z$. …

8: Bibliography K

… ►

ZEBEC: A mathematical software package for computing simple zeros of Bessel functions of real order and complex argument. Comput. Phys. Comm. 113 (2-3), pp. 220–238.

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Double-precision Fortran, maximum accuracy 25S.

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ISSN 0010-4655, Document, Code, ZentralBlatt

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9: 22.4 Periods, Poles, and Zeros

… ►Then: (a) In any lattice unit cell $\mathrm{p}\mathrm{q}(z,k)$ has a simple zero at $z=\text{p}$ and a simple pole at $z=\text{q}$. …

10: 23.2 Definitions and Periodic Properties

… ►The function $\sigma \left(z\right)$ is entire and odd, with simple zeros at the lattice points. …