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11—20 of 749 matching pages

11: 23.15 Definitions

… ►Also

\mathcal{A}

denotes a bilinear transformation on

\tau

, given by … ►A modular function

f(\tau)

is a function of

\tau

that is meromorphic in the half-plane

\Im\tau>0

, and has the property that for all

\mathcal{A}\in\mbox{SL}(2,\mathbb{Z})

, or for all

\mathcal{A}

belonging to a subgroup of SL

(2,\mathbb{Z})

, ►

23.15.5

f(\mathcal{A}\tau)=c_{\mathcal{A}}(c\tau+d)^{\ell}f(\tau),

\Im\tau>0

ⓘ

Symbols:: $\Im$ : imaginary part, $\tau$ : complex variable, $\mathcal{A}$ : bilinear transformation, $c$ : integer, $d$ : integer, $f(\tau)$ : modular function, $c_{\mathcal{A}}$ : constant and $\ell$ : level
Permalink:: http://dlmf.nist.gov/23.15.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §23.15(i), §23.15 and Ch.23

►where

c_{\mathcal{A}}

is a constant depending only on

\mathcal{A}

, and

\ell

(the level) is an integer or half an odd integer. … ►

23.15.7

J\left(\tau\right)=\frac{\left({\theta_{2}}^{8}\left(0,q\right)+{\theta_{3}}^{% 8}\left(0,q\right)+{\theta_{4}}^{8}\left(0,q\right)\right)^{3}}{54\left(\theta% _{1}'\left(0,q\right)\right)^{8}},

ⓘ

Defines:: $J\left(\NVar{\tau}\right)$ : Klein’s complete invariant
Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $q$ : nome and $\tau$ : complex variable
Permalink:: http://dlmf.nist.gov/23.15.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §23.15(ii), §23.15(ii), §23.15 and Ch.23

…

12: 26.10 Integer Partitions: Other Restrictions

… ►The set

\{n\geq 1\>|\>n\equiv\pm j\ \pmod{k}\}

is denoted by

A_{j,k}

. … ►Note that

p\left(\mathcal{D}^{\prime}3,n\right)\leq p\left(\mathcal{D}3,n\right)

, with strict inequality for

n\geq 9

. It is known that for

k>3

p\left(\mathcal{D}k,n\right)\geq p\left(\in\!A_{1,k+3},n\right)

, with strict inequality for

n

sufficiently large, provided that

k=2^{m}-1,m=3,4,5

, or

k\geq 32

; see Yee (2004). … ►where

I_{1}\left(x\right)

is the modified Bessel function (§10.25(ii)), and …The quantity

A_{k}(n)

is real-valued. …

13: 1.1 Special Notation

… ► ►►►►►►

$x, y$	real variables.
…
${\mathbf{A}}^{-1}$	inverse of the square matrix $\mathbf{A}$
…
$\det(\mathbf{A})$	determinant of the square matrix $\mathbf{A}$
$\operatorname{tr}(\mathbf{A})$	trace of the square matrix $\mathbf{A}$
$\operatorname{etr}\left(\mathbf{A}\right)$	exponential of $\operatorname{tr}(\mathbf{A})$
…

►In the physics, applied maths, and engineering literature a common alternative to

\overline{a}

a^{*}

a

being a complex number or a matrix; the Hermitian conjugate of

\mathbf{A}

is usually being denoted

\mathbf{A}^{{\dagger}}

14: 3.7 Ordinary Differential Equations

… ►Consideration will be limited to ordinary linear second-order differential equations … ►where

\mathbf{A}(\tau,z)

is the matrix … ►Let

\mathbf{A}_{P}

be the

(2P)\times(2P+2)

band matrix … ►If, for example,

\beta_{0}=\beta_{1}=0

, then on moving the contributions of

w(z_{0})

and

w(z_{P})

to the right-hand side of (3.7.13) the resulting system of equations is not tridiagonal, but can readily be made tridiagonal by annihilating the elements of

\mathbf{A}_{P}

that lie below the main diagonal and its two adjacent diagonals. … ►If

q(x)

C^{\infty}

on the closure of

(a,b)

, then the discretized form (3.7.13) of the differential equation can be used. …

15: Bibliography K

… ►

N. D. Kazarinoff (1988) Special functions and the Bieberbach conjecture. Amer. Math. Monthly 95 (8), pp. 689–696.

ⓘ

External Links:: ISSN 0002-9890, FullText, MathReview (S. L. Shukla), ZentralBlatt
Referenced by:: §16.23(iii).
Encodings:: BibTeX

… ►

M. K. Kerimov and S. L. Skorokhodov (1984b) Calculation of the complex zeros of the modified Bessel function of the second kind and its derivatives. Zh. Vychisl. Mat. i Mat. Fiz. 24 (8), pp. 1150–1163.

ⓘ

Notes:: English translation in U.S.S.R. Computational Math. and Math. Phys. 24(4), pp. 115–123
External Links:: ISSN 0044-4669, Document, MathReview (Walter Gautschi), ZentralBlatt
Referenced by:: §10.42, §10.74(vi), 3rd item.
Encodings:: BibTeX

… ►

T. H. Koornwinder (1977) The addition formula for Laguerre polynomials. SIAM J. Math. Anal. 8 (3), pp. 535–540.

ⓘ

External Links:: Document, MathReview (M. Mikolas), ZentralBlatt
Referenced by:: (18.14.8), §18.14(i), §18.17(ii), §18.18(ii).
Encodings:: BibTeX

… ►

B. G. Korenev (2002) Bessel Functions and their Applications. Analytical Methods and Special Functions, Vol. 8, Taylor & Francis Ltd., London-New York.

ⓘ

Notes:: Translated from the Russian by E. V. Pankratiev
External Links:: ISBN 0-415-28130-X, MathReview (L. Debnath), ZentralBlatt
Referenced by:: §10.73(i), §10.73(i), §10.73(i).
Encodings:: BibTeX

… ►

E. D. Krupnikov and K. S. Kölbig (1997) Some special cases of the generalized hypergeometric function

{}_{q+1}F_{q}

. J. Comput. Appl. Math. 78 (1), pp. 79–95.

ⓘ

External Links:: ISSN 0377-0427, Document, MathReview (Boro Döring), ZentralBlatt
Referenced by:: §16.7.
Encodings:: BibTeX

…

16: 3.3 Interpolation

… ►where

C

is a simple closed contour in

D

described in the positive rotational sense and enclosing the points

z,z_{1},z_{2},\dots,z_{n}

. … ►and

A_{k}^{n}

are the Lagrangian interpolation coefficients defined by … ►Let

c_{n}

be defined by … ►where

\omega_{n+1}(\zeta)

is given by (3.3.3), and

C

is a simple closed contour in

{D}

described in the positive rotational sense and enclosing

z_{0},z_{1},\dots,z_{n}

. … ►Then by using

x_{3}

in Newton’s interpolation formula, evaluating

\left[x_{0},x_{1},x_{2},x_{3}\right]f=-0.26608\;28233

and recomputing

f^{\prime}(x)

, another application of Newton’s rule with starting value

x_{3}

gives the approximation

x=2.33810\;7373

, with 8 correct digits. …

17: 27.2 Functions

… ►Functions in this section derive their properties from the fundamental theorem of arithmetic, which states that every integer

n>1

can be represented uniquely as a product of prime powers, …(

\nu\left(1\right)

is defined to be 0.) …It can be expressed as a sum over all primes

p\leq x

: … ►is the sum of the

\alpha

th powers of the divisors of

n

, where the exponent

\alpha

can be real or complex. Note that

\sigma_{0}\left(n\right)=d\left(n\right)

. …

18: 12.14 The Function $W\left(a,x\right)$

… ►the branch of

\operatorname{ph}

being zero when

a=0

and defined by continuity elsewhere. … ►Other expansions, involving

\cos\left(\tfrac{1}{4}x^{2}\right)

and

\sin\left(\tfrac{1}{4}x^{2}\right)

, can be obtained from (12.4.3) to (12.4.6) by replacing

a

-ia

and

z

xe^{\ifrac{\pi i}{4}}

; see Miller (1955, p. 80), and also (12.14.15) and (12.14.16). … ►Here

{\cal{A}}_{s}(t)

is as in §12.10(ii),

\sigma

is defined by … ►uniformly for

t\in[-1+\delta,1-\delta]

, with

\eta

given by (12.10.23) and

{\widetilde{\cal A}}_{s}(t)

given by (12.10.24). … ►uniformly for

t\in[-1+\delta,\infty)

, with

\zeta

\phi(\zeta)

A_{s}(\zeta)

, and

B_{s}(\zeta)

as in §12.10(vii). …

19: 28.31 Equations of Whittaker–Hill and Ince

… ►and constant values of

A, B, k

, and

c

, is called the Equation of Whittaker–Hill. … ►When

k^{2}<0

, we substitute … ►They are denoted by … ►They are real and distinct, and can be ordered so that

C_{p}^{m}(z,\xi)

and

S_{p}^{m}(z,\xi)

have precisely

m

zeros, all simple, in

0\leq z<\pi

. …ambiguities in sign being resolved by requiring

C_{p}^{m}(x,\xi)

and

{S_{p}^{m}}^{\prime}(x,\xi)

to be continuous functions of

x

and positive when

x=0

. …

20: 1.11 Zeros of Polynomials

… ►Set

z=w-\tfrac{1}{3}a

to reduce

f(z)=z^{3}+az^{2}+bz+c

g(w)=w^{3}+pw+q

, with

p=(3b-a^{2})/3

q=(2a^{3}-9ab+27c)/27

. … ►

f(z)=z^{3}-6z^{2}+6z-2

g(w)=w^{3}-6w-6

A=3\sqrt[3]{4}

B=3\sqrt[3]{2}

. … ►Resolvent cubic is

z^{3}+12z^{2}+20z+9=0

with roots

\theta_{1}=-1

\theta_{2}=-\tfrac{1}{2}(11+\sqrt{85})

\theta_{3}=-\tfrac{1}{2}(11-\sqrt{85})

, and

\sqrt{-\theta_{1}}=1

\sqrt{-\theta_{2}}=\tfrac{1}{2}(\sqrt{17}+\sqrt{5})

\sqrt{-\theta_{3}}=\tfrac{1}{2}(\sqrt{17}-\sqrt{5})

. … ►Let … ►Then

f(z)

, with

a_{n}\not=0

, is stable iff

a_{0}\not=0

;

D_{2k}>0

k=1,\dots,\left\lfloor\frac{1}{2}n\right\rfloor

;

\operatorname{sign}D_{2k+1}=\operatorname{sign}a_{0}

k=0,1,\dots,\left\lfloor\frac{1}{2}n-\frac{1}{2}\right\rfloor

11—20 of 749 matching pages

11: 23.15 Definitions

12: 26.10 Integer Partitions: Other Restrictions

13: 1.1 Special Notation

14: 3.7 Ordinary Differential Equations

15: Bibliography K

16: 3.3 Interpolation

17: 27.2 Functions

18: 12.14 The Function W ⁡ ( a , x )

19: 28.31 Equations of Whittaker–Hill and Ince

20: 1.11 Zeros of Polynomials

18: 12.14 The Function $W\left(a,x\right)$