analytic functions

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11: 22.17 Moduli Outside the Interval [0,1]

… ►

§22.17(ii) Complex Moduli

…

12: 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. … ►

Analytic Functions

… ►Suppose that

a_{n}(z)

are analytic functions in

D

. …

13: 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. …

14: 1.13 Differential Equations

… ►Assume that in the equation …

u

and

z

belong to domains

U

and

D

respectively, the coefficients

f(u,z)

and

g(u,z)

are continuous functions of both variables, and for each fixed

u

(fixed

z

) the two functions are analytic in

z

(in

u

). Suppose also that at (a fixed)

z_{0}\in D

w

and

\ifrac{\partial w}{\partial z}

are analytic functions of

u

. Then at each

z\in D

w

\ifrac{\partial w}{\partial z}

and

\ifrac{{\partial}^{2}w}{{\partial z}^{2}}

are analytic functions of

u

. … ►The inhomogeneous (or nonhomogeneous) equation …

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)

. …

16: 14.21 Definitions and Basic Properties

… ► …

17: 14.24 Analytic Continuation

§14.24 Analytic Continuation

…

18: 10.34 Analytic Continuation

§10.34 Analytic Continuation

…

19: 10.11 Analytic Continuation

§10.11 Analytic Continuation

…

20: 2.7 Differential Equations

… ►

2.7.1

\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+f(z)\frac{\mathrm{d}w}{\mathrm{d}z% }+g(z)w=0

ⓘ

Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $w$ : 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
Encodings:: pMML, png, TeX
See also:: 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. …