# zeros of analytic functions

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##### 1: 3.8 Nonlinear Equations
###### §3.8(v) Zeros of AnalyticFunctions
Newton’s rule is the most frequently used iterative process for accurate computation of real or complex zeros of analytic functions $f(z)$. …
###### §3.8(vi) Conditioning of Zeros
Corresponding numerical factors in this example for other zeros and other values of $j$ are obtained in Gautschi (1984, §4). …
##### 2: 4.14 Definitions and Periodicity
4.14.7 $\cot z=\frac{\cos z}{\sin z}=\frac{1}{\tan z}.$
##### 3: 5.2 Definitions
5.2.1 $\Gamma\left(z\right)=\int_{0}^{\infty}e^{-t}t^{z-1}\,\mathrm{d}t,$ $\Re z>0$.
##### 4: 1.10 Functions of a Complex Variable
An analytic function $f(z)$ has a zero of order (or multiplicity) $m$ ($\geq\!1$) at $z_{0}$ if the first nonzero coefficient in its Taylor series at $z_{0}$ is that of $(z-z_{0})^{m}$. … … then the product $\prod^{\infty}_{n=1}(1+a_{n}(z))$ converges uniformly to an analytic function $p(z)$ in $D$, and $p(z)=0$ only when at least one of the factors $1+a_{n}(z)$ is zero in $D$. …
##### 7: Bibliography C
• J. A. Cochran (1966a) The analyticity of cross-product Bessel function zeros. Proc. Cambridge Philos. Soc. 62, pp. 215–226.
##### 9: 23.2 Definitions and Periodic Properties
###### §23.2(ii) Weierstrass Elliptic Functions
The function $\sigma\left(z\right)$ is entire and odd, with simple zeros at the lattice points. … …
##### 10: 2.4 Contour Integrals
Now assume that $c>0$ and we are given a function $Q(z)$ that is both analytic and has the expansion … Assume that $p(t)$ and $q(t)$ are analytic on an open domain $\mathbf{T}$ that contains $\mathscr{P}$, with the possible exceptions of $t=a$ and $t=b$. …
• (a)

In a neighborhood of $a$

with $\Re\lambda>0$, $\mu>0$, $p_{0}\neq 0$, and the branches of $(t-a)^{\lambda}$ and $(t-a)^{\mu}$ continuous and constructed with $\operatorname{ph}\left(t-a\right)\to\omega$ as $t\to a$ along $\mathscr{P}$.

• in which $z$ is a large real or complex parameter, $p(\alpha,t)$ and $q(\alpha,t)$ are analytic functions of $t$ and continuous in $t$ and a second parameter $\alpha$. … The function $f(\alpha,w)$ is analytic at $w=w_{1}(\alpha)$ and $w=w_{2}(\alpha)$ when $\alpha\neq\widehat{\alpha}$, and at the confluence of these points when $\alpha=\widehat{\alpha}$. …