analytic functions

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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 $ℂ\setminus (-\mathrm{\infty },0]$ and real-valued when $z$ ranges over the positive real numbers. … ►

§4.2(iii) The Exponential Function

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4.2.27 $$z^a=\underset{n\text{ times}}{\underbrace{z\cdot z\mathrm{\cdots }z}}=1/z^{-a}.$$

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Defines:

$a=n$: integer (locally)

Symbols:

$n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.2.E27

Encodings:

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Annotations for §4.2(iv), §4.2(iv), §4.2 and Ch.4

… ►This is an analytic function of $z$ on $ℂ\setminus (-\mathrm{\infty },0]$, and is two-valued and discontinuous on the cut shown in Figure 4.2.1, unless $a\in \mathbb{Z}$. …

12: 22.17 Moduli Outside the Interval [0,1]

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§22.17(ii) Complex Moduli

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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 $ℂ$ with at most one exception. … ►

Analytic Functions

… ►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 $dw/dz$, and is locally analytic in $z$, that is, analytic except for isolated singularities in $ℂ$. … ►in which $a(z)$, $b(z)$, $c(z)$, $d(z)$, and $\varphi (z)$ are locally analytic functions. …

15: 1.9 Calculus of a Complex Variable

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Analyticity

… ►A function analytic at every point of $ℂ$ is said to be entire. … ► … ►Inside the circle the sum of the series is an analytic function $f(z)$. …

16: 14.21 Definitions and Basic Properties

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17: 14.24 Analytic Continuation

§14.24 Analytic Continuation

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18: 10.34 Analytic Continuation

§10.34 Analytic Continuation

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19: 10.11 Analytic Continuation

§10.11 Analytic Continuation

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20: 2.7 Differential Equations

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2.7.1 $$\frac{d^{2}w}{{dz}^2}+f(z)\frac{dw}{dz}+g(z)w=0$$

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Symbols:

$\frac{df}{dx}$: derivative, $w$: DE solution, $f(z)$: analytic function and $g(z)$: analytic function

Referenced by:

§2.11(v), §2.7(i), §2.7(i)

Permalink:

http://dlmf.nist.gov/2.7.E1

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Annotations for §2.7(i), §2.7 and Ch.2

… ►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. …