# analytic functions

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## 11—20 of 142 matching pages

##### 11: 4.2 Definitions
This is a multivalued function of $z$ with branch point at $z=0$. … $\ln z$ is a single-valued analytic function on $\mathbb{C}\setminus(-\infty,0]$ and real-valued when $z$ ranges over the positive real numbers. …
###### §4.2(iii) The Exponential Function
4.2.27 $z^{a}=\underbrace{z\cdot z\cdots z}_{n\text{ times}}=1/z^{-a}.$
This is an analytic function of $z$ on $\mathbb{C}\setminus(-\infty,0]$, and is two-valued and discontinuous on the cut shown in Figure 4.2.1, unless $a\in\mathbb{Z}$. …
##### 13: 1.10 Functions of a Complex Variable
In any neighborhood of an isolated essential singularity, however small, an analytic function assumes every value in $\mathbb{C}$ with at most one exception. …
###### AnalyticFunctions
Suppose that $a_{n}(z)$ are analytic functions in $D$. …
##### 14: 32.2 Differential Equations
be a nonlinear second-order differential equation in which $F$ is a rational function of $w$ and $\ifrac{\mathrm{d}w}{\mathrm{d}z}$, and is locally analytic in $z$, that is, analytic except for isolated singularities in $\mathbb{C}$. … in which $a(z)$, $b(z)$, $c(z)$, $d(z)$, and $\phi(z)$ are locally analytic functions. …
##### 15: 1.9 Calculus of a Complex Variable
###### Analyticity
A function analytic at every point of $\mathbb{C}$ is said to be entire. … Inside the circle the sum of the series is an analytic function $f(z)$. …
##### 20: 2.7 Differential Equations
All solutions are analytic at an ordinary point, and their Taylor-series expansions are found by equating coefficients. … In a finite or infinite interval $(a_{1},a_{2})$ let $f(x)$ be real, positive, and twice-continuously differentiable, and $g(x)$ be continuous. …