DLMF: Untitled Document

… ►If all solutions of (28.2.1) are bounded when

x\to\pm\infty

along the real axis, then the corresponding pair of parameters

(a,q)

is called stable. … ►However, if

\Im\nu\neq 0

, then

(a,q)

always comprises an unstable pair. … ►For real

a

and

q

(\neq 0)

the stable regions are the open regions indicated in color in Figure 28.17.1. …

… ►The function

(1-t^{2})^{1/2}

assumes its principal value when

t\in(-1,1)

; elsewhere on the integration paths the branch is determined by continuity. … ►

4.23.19

\operatorname{arcsin}z=-i\ln\left((1-z^{2})^{1/2}+iz\right),

z\in\mathbb{C}\setminus(-\infty,-1)\cup(1,\infty)

;

ⓘ

Symbols:: $\mathbb{C}$ : complex plane, $\in$ : element of, $\mathrm{i}$ : imaginary unit, $\operatorname{arcsin}\NVar{z}$ : arcsine function, $\ln\NVar{z}$ : principal branch of logarithm function, $(\NVar{a},\NVar{b})$ : open interval, $\setminus$ : set subtraction, $\cup$ : union and $z$ : complex variable
Referenced by:: §4.23(iv)
Permalink:: http://dlmf.nist.gov/4.23.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §4.23(iv), §4.23(iv), §4.23 and Ch.4

… ►

4.23.22

\operatorname{arccos}z=\tfrac{1}{2}\pi+i\ln\left((1-z^{2})^{1/2}+iz\right),

z\in\mathbb{C}\setminus(-\infty,-1)\cup(1,\infty)

;

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathbb{C}$ : complex plane, $\in$ : element of, $\mathrm{i}$ : imaginary unit, $\operatorname{arccos}\NVar{z}$ : arccosine function, $\ln\NVar{z}$ : principal branch of logarithm function, $(\NVar{a},\NVar{b})$ : open interval, $\setminus$ : set subtraction, $\cup$ : union and $z$ : complex variable
Referenced by:: §4.23(iv)
Permalink:: http://dlmf.nist.gov/4.23.E22
Encodings:: pMML, png, TeX
See also:: Annotations for §4.23(iv), §4.23(iv), §4.23 and Ch.4

… ►

4.23.26

\operatorname{arctan}z=\frac{i}{2}\ln\left(\frac{i+z}{i-z}\right),

z/i\in\mathbb{C}\setminus(-\infty,-1]\cup[1,\infty)

;

ⓘ

Symbols:: $[\NVar{a},\NVar{b})$ : half-closed interval, $\mathbb{C}$ : complex plane, $\in$ : element of, $\mathrm{i}$ : imaginary unit, $\operatorname{arctan}\NVar{z}$ : arctangent function, $\ln\NVar{z}$ : principal branch of logarithm function, $(\NVar{a},\NVar{b}]$ : half-closed interval, $\setminus$ : set subtraction, $\cup$ : union and $z$ : complex variable
Referenced by:: §4.23(iv)
Permalink:: http://dlmf.nist.gov/4.23.E26
Encodings:: pMML, png, TeX
See also:: Annotations for §4.23(iv), §4.23(iv), §4.23 and Ch.4

… ►where

z=x+\mathrm{i}y

and

\pm z\notin(1,\infty)

in (4.23.34) and (4.23.35), and

\left|z\right|<1

in (4.23.36). …

… ►then there are exactly

\genfrac{(}{)}{0.0pt}{}{n+N-2}{N-2}

polynomials

S(z)

, each of which corresponds to each of the

\genfrac{(}{)}{0.0pt}{}{n+N-2}{N-2}

ways of distributing its

n

zeros among

N-1

intervals

(a_{j},a_{j+1})

,

j=1,2,\dots,N-1

. … ►If the exponent and singularity parameters satisfy (31.15.5)–(31.15.6), then for every multi-index

\mathbf{m}=(m_{1},m_{2},\dots,m_{N-1})

, where each

m_{j}

is a nonnegative integer, there is a unique Stieltjes polynomial with

m_{j}

zeros in the open interval

(a_{j},a_{j+1})

for each

j=1,2,\dots,N-1

. … ►

31.15.8

S_{\mathbf{m}}(z_{1})S_{\mathbf{m}}(z_{2})\cdots S_{\mathbf{m}}(z_{N-1}),

z_{j}\in(a_{j},a_{j+1})

,

ⓘ

Symbols:: $\in$ : element of, $(\NVar{a},\NVar{b})$ : open interval, $z$ : complex variable, $j$ : nonnegative integer, $a$ : complex parameter, $N+1$ : number of singularities and $S_{\mathbf{m}}(z)$ : Stieltjes polynomial
Referenced by:: §31.15(iii)
Permalink:: http://dlmf.nist.gov/31.15.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §31.15(iii), §31.15 and Ch.31

►

31.15.9

S_{\mathbf{l}}(z_{1})S_{\mathbf{l}}(z_{2})\cdots S_{\mathbf{l}}(z_{N-1}),

z_{j}\in(a_{j},a_{j+1})

,

ⓘ

Symbols:: $\in$ : element of, $(\NVar{a},\NVar{b})$ : open interval, $z$ : complex variable, $j$ : nonnegative integer, $a$ : complex parameter, $N+1$ : number of singularities and $S_{\mathbf{m}}(z)$ : Stieltjes polynomial
Permalink:: http://dlmf.nist.gov/31.15.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §31.15(iii), §31.15 and Ch.31

… ►

31.15.10

Q=(a_{1},a_{2})\times(a_{2},a_{3})\times\cdots\times(a_{N-1},a_{N}),

ⓘ

Defines:: $Q$ : set (locally)
Symbols:: $(\NVar{a},\NVar{b})$ : open interval, $a$ : complex parameter and $N+1$ : number of singularities
Permalink:: http://dlmf.nist.gov/31.15.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §31.15(iii), §31.15 and Ch.31

…

… ►

Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, standardizations, leading coefficients, and parameter constraints. …

► ►►►►

Name	$p_{n}(x)$	$(a,b)$	$w(x)$	$h_{n}$	$k_{n}$	$\ifrac{\tilde{k}_{n}}{k_{n}}$
…
Chebyshev of second kind	$U_{n}\left(x\right)$	$(-1,1)$	$(1-x^{2})^{\frac{1}{2}}$	$\tfrac{1}{2}\pi$	$2^{n}$	$0$
Chebyshev of third kind	$V_{n}\left(x\right)$	$(-1,1)$	$(1-x)^{-\frac{1}{2}}(1+x)^{\frac{1}{2}}$	$\pi$	$2^{n}$	$-\tfrac{1}{2}$
…

► … ►For

-1-\beta>\alpha>-1

a finite system of Jacobi polynomials

P^{(\alpha,\beta)}_{n}\left(x\right)

is orthogonal on

(1,\infty)

with weight function

w(x)=(x-1)^{\alpha}(x+1)^{\beta}

. For

\nu\in\mathbb{R}

and

N>-\tfrac{1}{2}

a finite system of Jacobi polynomials

P^{(-N-1+\mathrm{i}\nu,-N-1-\mathrm{i}\nu)}_{n}\left(\mathrm{i}x\right)

(called pseudo Jacobi polynomials or Routh–Romanovski polynomials) is orthogonal on

(-\infty,\infty)

with

w(x)=\left(1+x^{2}\right)^{-N-1}{\mathrm{e}}^{2\nu\operatorname{arctan}x}

. …

… ► ►►►►

$j,k,\ell,m,n$	integers, nonnegative unless stated otherwise.
…
$(k,m)$	greatest common divisor of $k, m$ .
$(k,m)=1$	$k$ and $m$ relatively prime.

…

… ►Points

P=(x,y)

on the curve can be parametrized by

x=\wp\left(z;g_{2},g_{3}\right)

,

2y=\wp'\left(z;g_{2},g_{3}\right)

, where

g_{2}=-4a

and

g_{3}=-4b

: in this case we write

P=P(z)

. The curve

C

is made into an abelian group (Macdonald (1968, Chapter 5)) by defining the zero element

o=(0,1,0)

as the point at infinity, the negative of

P=(x,y)

by

-P=(x,-y)

, and generally

P_{1}+P_{2}+P_{3}=0

on the curve iff the points

P_{1}

,

P_{2}

,

P_{3}

are collinear. … ►In terms of

(x,y)

the addition law can be expressed

(x,y)+o=(x,y)

,

(x,y)+(x,-y)=o

; otherwise

(x_{1},y_{1})+(x_{2},y_{2})=(x_{3},y_{3})

, where … ►

K

always has the form

T\times{\mathbb{Z}}^{r}

(Mordell’s Theorem: Silverman and Tate (1992, Chapter 3, §5)); the determination of

r

, the rank of

K

, raises questions of great difficulty, many of which are still open. …To determine

T

, we make use of the fact that if

(x,y)\in T

then

y^{2}

must be a divisor of

\Delta

; hence there are only a finite number of possibilities for

y

. …

… ►The Bernstein–Szegő polynomials

\{p_{n}(x)\}

,

n=0,1,\dots

, are orthogonal on

(-1,1)

with respect to three types of weight function:

(1-x^{2})^{-\frac{1}{2}}(\rho(x))^{-1}

,

(1-x^{2})^{\frac{1}{2}}(\rho(x))^{-1}

,

(1-x)^{\frac{1}{2}}(1+x)^{-\frac{1}{2}}(\rho(x))^{-1}

. …

… ►

•

Schonfelder (1978) gives coefficients of Chebyshev expansions for $x^{-1}\operatorname{erf}x$ on $0\leq x\leq 2$ , for $xe^{x^{2}}\operatorname{erfc}x$ on $[2,\infty)$ , and for $e^{x^{2}}\operatorname{erfc}x$ on $[0,\infty)$ (30D).

►

•

Shepherd and Laframboise (1981) gives coefficients of Chebyshev series for $(1+2x)e^{x^{2}}\operatorname{erfc}x$ on $(0,\infty)$ (22D).

…

… ► ►►►►

$\mathbb{L}$	lattice in $\mathbb{C}$ .
…
$[a,b]$ or $(a,b)$	closed, or open, straight-line segment joining $a$ and $b$ , whether or not $a$ and $b$ are real.
…
$G\times H$	Cartesian product of groups $G$ and $H$ , that is, the set of all pairs of elements $(g,h)$ with group operation $(g_{1},h_{1})+(g_{2},h_{2})=(g_{1}+g_{2},h_{1}+h_{2})$ .

…

… ► ►►►

$\mathbb{C}$	complex plane (excluding infinity).
…
$(a,b)$	open interval in $\mathbb{R}$ , or open straight-line segment joining $a$ and $b$ in $\mathbb{C}$ .
…

► ►►►

$(a,b]$ or $[a,b)$	half-closed intervals.
…
$[a_{j,k}]$ or $[a_{jk}]$	matrix with $(j,k)$ th element $a_{j,k}$ or $a_{jk}$ .
…

…

open

11—20 of 195 matching pages

11: 28.17 Stability as $x\to\pm\infty$

12: 4.23 Inverse Trigonometric Functions

13: 31.15 Stieltjes Polynomials

14: 18.3 Definitions

15: 24.1 Special Notation

16: 23.20 Mathematical Applications

17: 18.31 Bernstein–Szegő Polynomials

18: 7.24 Approximations

19: 23.1 Special Notation

20: Mathematical Introduction