DLMF: Untitled Document

… ►The mathematical content of the NIST Handbook of Mathematical Functions has been produced over a ten-year period. … ►First, the editors instituted a validation process for the whole technical content of each chapter. … ►Secondly, as described in the Preface, a Web version (the NIST DLMF) is also available. … ►These include, for example, multivalued functions of complex variables, for which new definitions of branch points and principal values are supplied (§§1.10(vi), 4.2(i)); the Dirac delta (or delta function), which is introduced in a more readily comprehensible way for mathematicians (§1.17); numerically satisfactory solutions of differential and difference equations (§§2.7(iv), 2.9(i)); and numerical analysis for complex variables (Chapter 3). … ►Special functions with a complex variable are depicted as colored 3D surfaces in a similar way to functions of two real variables, but with the vertical height corresponding to the modulus (absolute value) of the function. …

… ►The terminology DVR arises as an otherwise continuous variable, such as the co-ordinate

x

, is replaced by its values at a finite set of zeros of appropriate OP’s resulting in expansions using functions localized at these points. …

… ►Inside the turning points, that is, when

\rho<\rho_{\operatorname{tp}}\left(\eta,\ell\right)

, there can be a loss of precision by a factor of approximately

|G_{\ell}|^{2}

. … ►WKBJ approximations (§2.7(iii)) for

\rho>\rho_{\operatorname{tp}}\left(\eta,\ell\right)

are presented in Hull and Breit (1959) and Seaton and Peach (1962: in Eq. (12)

(\rho-c)/c

should be

(\rho-c)/\rho

). A set of consistent second-order WKBJ formulas is given by Burgess (1963: in Eq. … ►Hull and Breit (1959) and Barnett (1981b) give WKBJ approximations for

F_{0}

and

G_{0}

in the region inside the turning point:

\rho<\rho_{\operatorname{tp}}\left(\eta,\ell\right)

.

… ►This equation has regular singularities at the points

2pK+(2q+1)\mathrm{i}{K^{\prime}}

, where

p,q\in\mathbb{Z}

, and

K

,

{K^{\prime}}

are the complete elliptic integrals of the first kind with moduli

k

,

k^{\prime}(=(1-k^{2})^{1/2})

, respectively; see §19.2(ii). In general, at each singularity each solution of (29.2.1) has a branch point (§2.7(i)). … ►

\ifrac{(e_{2}-e_{3})}{(e_{1}-e_{3})}=k^{2}.

… ►

29.2.8

\eta=(e_{1}-e_{3})^{-1/2}(z-\mathrm{i}{K^{\prime}}),

ⓘ

Defines:: $\eta$ (locally)
Symbols:: ${K^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $\mathrm{i}$ : imaginary unit, $z$ : complex variable, $k$ : real parameter and $e_{j}$ : constants
Permalink:: http://dlmf.nist.gov/29.2.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §29.2(ii), §29.2 and Ch.29

… ►Equation (29.2.10) is a special case of Heun’s equation (31.2.1). …

… ►

mpmath (free python library)

ⓘ

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Encodings:: BibTeX

…

… ►With the choice

r=k

(which is crucial when

k

is large because of numerical cancellation) the integrand equals

e^{k}

at the dominant points

\theta=0,2\pi

, and in combination with the factor

k^{-k}

in front of the integral sign this gives a rough approximation to

1/k!

. …

… ►As

\omega

runs from

0

to

+\infty

, with

b

and

f

fixed, the point

(q,a)

moves from

\infty

to

0

along the ray

\mathcal{L}

given by the part of the line

a=(2b/f)q

that lies in the first quadrant of the

(q,a)

-plane. …

… ►The closure of the set of points where

\phi\not=0

is called the support of

\phi

. If the support of

\phi

is a compact set (§1.9(vii)), then

\phi

is called a function of compact support. … ►A sequence

\{\phi_{n}\}

of test functions converges to a test function

\phi

if the support of every

\phi_{n}

is contained in a fixed compact set

K

and as

n\to\infty

the sequence

\{\phi_{n}^{(k)}\}

converges uniformly on

K

to

\phi^{(k)}

for

k=0,1,2,\dots

. … ►If

k=(k_{1},\dots,k_{n})

is a multi-index and

x=(x_{1},\dots,x_{n})\in{\mathbb{R}}^{n}

, then we write

x^{k}=x_{1}^{k_{1}}\cdots x_{n}^{k_{n}}

and

\phi^{(k)}(x)=\,{\partial}^{k}\phi/(\,\partial x_{1}^{k_{1}}\cdots\,\partial x% _{n}^{k_{n}})

. … ►Here

\boldsymbol{{\alpha}}

ranges over a finite set of multi-indices,

P(\mathbf{x})

is a multivariate polynomial, and

P(\mathbf{D})

is a partial differential operator. …

… ►If

f(x)

is continuous at each point

c\in(a,b)

, then

f(x)

is continuous on the interval

(a,b)

and we write

f\in C(a,b)

. … ►where the supremum is over all sets of points

x_{0}<x_{1}<\cdots<x_{n}

in the closure of

(a,b)

, that is,

(a,b)

with

a, b

added when they are finite. …

… ►

See accompanying text — Figure 22.3.2: $k=0.7$ , $-3K\leq x\leq 3K$ , $K=1.8456\dots$ . For $\operatorname{cn}\left(x,k\right)$ the curve for $k=1/\sqrt{2}=0.70710\dots$ is a boundary between the curves that have an inflection point in the interval $0\leq x\leq 2K\left(k\right)$ , and its translates, and those that do not; see Walker (1996, p. 146). Magnify

…

on a point set

21—30 of 113 matching pages

21: Mathematical Introduction

22: 18.38 Mathematical Applications

23: 33.23 Methods of Computation

24: 29.2 Differential Equations

25: Bibliography M

26: 3.4 Differentiation

27: 28.33 Physical Applications

28: 1.16 Distributions

29: 1.4 Calculus of One Variable

30: 22.3 Graphics