# 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 $\tfrac{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 $z=-n$. $1/\Gamma\left(z\right)$ is entire, with simple zeros at $z=-n$. …
##### 3: 3.8 Nonlinear Equations
###### §3.8 Nonlinear Equations
If $f(z_{0})=0$ and $f^{\prime}(z_{0})\neq 0$, 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 …
##### 4: 7.13 Zeros
$\operatorname{erf}z$ has a simple zero at $z=0$, and in the first quadrant of $\mathbb{C}$ there is an infinite set of zeros $z_{n}=x_{n}+iy_{n}$, $n=1,2,3,\dots$, 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 $\left|g(z)\right|<\left|f(z)\right|$ 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 $\Re\left(zp(t)\right)$ is a simple zero of $p^{\prime}(t)$. … Suppose that on the integration path $\mathscr{P}$ there are two simple zeros of $\ifrac{\partial p(\alpha,t)}{\partial t}$ 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
• P. Kravanja, O. Ragos, M. N. Vrahatis, and F. A. Zafiropoulos (1998) 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.
• ##### 9: 22.4 Periods, Poles, and Zeros
Then: (a) In any lattice unit cell $\operatorname{pq}\left(z,k\right)$ has a simple zero at $z=\mbox{p}$ and a simple pole at $z=\mbox{q}$. …
##### 10: 23.2 Definitions and Periodic Properties
The function $\sigma\left(z\right)$ is entire and odd, with simple zeros at the lattice points. …