.%E6%A2%85%E8%A5%BF%E5%92%8Cc%E7%BD%97%E7%AC%AC%E4%B8%80%E6%AC%A1%E4%B8%96%E7%95%8C%E6%9D%AF%E3%80%8E%E7%BD%91%E5%9D%80%3Amxsty.cc%E3%80%8F.%E6%97%B6%E4%B8%96%E7%95%8C%E6%9D%AF%E8%B5%9B%E7%A8%8B-m6q3s2-qmymcyumu.com

♦ 1—10 of 736 matching pages ♦

Did you mean .%E6%A2%85%E8%A5%BF%E5%92%8Cc%E7%BD%97%E7%AC%AC%E4%B8%80%E6%AC%A1%E4%B8%96%E7%95%8C%E6%9D%AF%E3%80%8E%E7%BD%91%E5%9D%80%3Amxsty.cc%E3%80%8F.%E6%97%B6%E4%B8%96%E7%95%8C%E6%9D%AF%E8%B5%9B%E7%A8%8B-m6q3s2-commemor ?

(0.044 seconds)

1—10 of 736 matching pages

1: 9.13 Generalized Airy Functions

… ►Swanson and Headley (1967) define independent solutions

A_{n}\left(z\right)

and

B_{n}\left(z\right)

of (9.13.1) by …When

n=1

A_{n}\left(z\right)

and

B_{n}\left(z\right)

become

\operatorname{Ai}\left(z\right)

and

\operatorname{Bi}\left(z\right)

, respectively. ►Properties of

A_{n}\left(z\right)

and

B_{n}\left(z\right)

follow from the corresponding properties of the modified Bessel functions. … ►The distribution in

\mathbb{C}

and asymptotic properties of the zeros of

A_{n}\left(z\right)

A_{n}'\left(z\right)

B_{n}\left(z\right)

, and

B_{n}'\left(z\right)

are investigated in Swanson and Headley (1967) and Headley and Barwell (1975). … ►where

m=3,4,5,\ldots.

For real variables the solutions of (9.13.13) are denoted by

U_{m}(t)

U_{m}(-t)

when

m

is even, and by

V_{m}(t)

\overline{V}_{m}(t)

when

m

is odd. …

2: Joyce E. Conlon

… ►From 1980–85 she worked as a computer programmer for the Defense Mapping Agency. …

3: 19.32 Conformal Map onto a Rectangle

… ►

19.32.1

z(p)=R_{F}\left(p-x_{1},p-x_{2},p-x_{3}\right),

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind and $z(p)$ : function
Permalink:: http://dlmf.nist.gov/19.32.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §19.32 and Ch.19

►with

x_{1},x_{2},x_{3}

real constants, has differential ►

19.32.2

\,\mathrm{d}z=-\frac{1}{2}\left(\prod_{j=1}^{3}(p-x_{j})^{-1/2}\right)\,% \mathrm{d}p,

\Im p>0

;

0<\operatorname{ph}\left(p-x_{j}\right)<\pi

j=1,2,3

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\Im$ : imaginary part, $\operatorname{ph}$ : phase and $z(p)$ : function
Referenced by:: §19.32
Permalink:: http://dlmf.nist.gov/19.32.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §19.32 and Ch.19

… ►As

p

proceeds along the entire real axis with the upper half-plane on the right,

z

describes the rectangle in the clockwise direction; hence

z(x_{3})

is negative imaginary. ►For further connections between elliptic integrals and conformal maps, see Bowman (1953, pp. 44–85).

4: Bibliography Y

… ►

A. I. Yablonskiĭ (1959) On rational solutions of the second Painlevé equation. Vesti Akad. Navuk. BSSR Ser. Fiz. Tkh. Nauk. 3, pp. 30–35 (Russian).

ⓘ

Referenced by:: §32.8(ii).
Encodings:: BibTeX

… ►

K. Yang and M. de Llano (1989) Simple Variational Proof That Any Two-Dimensional Potential Well Supports at Least One Bound State. American Journal of Physics 57 (1), pp. 85–86.

ⓘ

External Links:: Document
Referenced by:: §1.18(viii).
Encodings:: BibTeX

…

5: 19.37 Tables

… ►

Functions $K\left(k\right)$ and $E\left(k\right)$

►Tabulated for

k^{2}=0(.01)1

to 6D by Byrd and Friedman (1971), to 15D for

K\left(k\right)

and 9D for

E\left(k\right)

by Abramowitz and Stegun (1964, Chapter 17), and to 10D by Fettis and Caslin (1964). … ►

Functions $F\left(\phi,k\right)$ and $E\left(\phi,k\right)$

… ►Tabulated for

\phi=0(5^{\circ})90^{\circ}

\operatorname{arcsin}k=0(1^{\circ})90^{\circ}

to 6D by Byrd and Friedman (1971), for

\phi=0(5^{\circ})90^{\circ}

\operatorname{arcsin}k=0(2^{\circ})90^{\circ}

and

5^{\circ}(10^{\circ})85^{\circ}

to 8D by Abramowitz and Stegun (1964, Chapter 17), and for

\phi=0(10^{\circ})90^{\circ}

\operatorname{arcsin}k=0(5^{\circ})90^{\circ}

to 9D by Zhang and Jin (1996, pp. 674–675). … ►Tabulated for

\phi=5^{\circ}(5^{\circ})80^{\circ}(2.5^{\circ})90^{\circ}

\alpha^{2}=-1(.1)-0.1,0.1(.1)1

k^{2}=0(.05)0.9(.02)1

to 10D by Fettis and Caslin (1964) (and warns of inaccuracies in Selfridge and Maxfield (1958) and Paxton and Rollin (1959)). …

6: Bibliography

… ►

C. Adiga, B. C. Berndt, S. Bhargava, and G. N. Watson (1985) Chapter

16

of Ramanujan’s second notebook: Theta-functions and

q

-series. Mem. Amer. Math. Soc. 53 (315), pp. v+85.

ⓘ

External Links:: ISSN 0065-9266, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §20.11(ii).
Encodings:: BibTeX

… ►

G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen (1992b) Hypergeometric Functions and Elliptic Integrals. In Current Topics in Analytic Function Theory, H. M. Srivastava and S. Owa (Eds.), pp. 48–85.

ⓘ

External Links:: MathReview (Bruce C. Berndt), ZentralBlatt
Referenced by:: §19.9(i).
Encodings:: BibTeX

… ►

G. E. Andrews, R. A. Askey, B. C. Berndt, and R. A. Rankin (Eds.) (1988) Ramanujan Revisited. Academic Press Inc., Boston, MA.

ⓘ

External Links:: ISBN 0-12-058560-X, ISBN 0-12-058560-X, MathReview, ZentralBlatt
Referenced by:: §20.11(ii), Profile Richard A. Askey.
Encodings:: BibTeX

… ►

T. M. Apostol and T. H. Vu (1984) Dirichlet series related to the Riemann zeta function. J. Number Theory 19 (1), pp. 85–102.

ⓘ

External Links:: ISSN 0022-314X, Document, MathReview (Kai Man Tsang), ZentralBlatt
Referenced by:: (25.16.10), (25.16.11), (25.16.12), (25.16.14), (25.16.15), (25.16.5), (25.16.6), (25.16.7), (25.16.8), (25.16.9), §25.16(ii), §25.16(ii), §25.16.
Encodings:: BibTeX

… ►

J. Avron and B. Simon (1982) Singular Continuous Spectrum for a Class of Almost Periodic Jacobi Matrices. Bulletin of the American Mathematical Society 6 (1), pp. 81–85.

ⓘ

External Links:: MathReview Entry
Referenced by:: §1.18(vii).
Encodings:: BibTeX

…

7: 26.6 Other Lattice Path Numbers

… ►

Delannoy Number $D(m,n)$

►

D(m,n)

is the number of paths from

(0,0)

(m,n)

that are composed of directed line segments of the form

(1,0)

(0,1)

, or

(1,1)

. ►

26.6.1

D(m,n)=\sum_{k=0}^{n}\genfrac{(}{)}{0.0pt}{}{n}{k}\genfrac{(}{)}{0.0pt}{}{m+n-% k}{n}=\sum_{k=0}^{n}2^{k}\genfrac{(}{)}{0.0pt}{}{m}{k}\genfrac{(}{)}{0.0pt}{}{% n}{k}.

ⓘ

Defines:: $D(m,n)$ : Delannoy number (locally)
Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : nonnegative integer, $m$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.6.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §26.6(i), §26.6(i), §26.6 and Ch.26

… ►

Table 26.6.1: Delannoy numbers

D(m,n)

► ►►

$m$	$n$
$m$	…

► … ►

26.6.4

r(n)=D(n,n)-D(n+1,n-1),

n\geq 1

ⓘ

Defines:: $r(n)$ : Schröder number (locally)
Symbols:: $n$ : nonnegative integer and $D(m,n)$ : Delannoy number
Referenced by:: §26.6(i)
Permalink:: http://dlmf.nist.gov/26.6.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §26.6(i), §26.6(i), §26.6 and Ch.26

…

8: 3.4 Differentiation

… ►The

B_{k}^{n}

are the differentiated Lagrangian interpolation coefficients: … ►

B_{1}^{7}=\tfrac{1}{144}(144+216t-264t^{2}-156t^{3}+85t^{4}+18t^{5}-7t^{6}),

… ►

B_{3}^{7}=\tfrac{1}{720}(48+8t-192t^{2}-20t^{3}+85t^{4}+6t^{5}-7t^{6}),

… ►where

C

is a simple closed contour described in the positive rotational sense such that

C

and its interior lie in the domain of analyticity of

f

, and

x_{0}

is interior to

C

. Taking

C

to be a circle of radius

r

centered at

x_{0}

, we obtain …

9: 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

. … ►Addition of

-\frac{1}{3}a

to each of these roots gives the roots of

f(z)=0

. … ►

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

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

A=3\sqrt[3]{4}

B=3\sqrt[3]{2}

. Roots of

f(z)=0

are

2+\sqrt[3]{4}+\sqrt[3]{2}

2+\sqrt[3]{4}\rho+\sqrt[3]{2}\rho^{2}

2+\sqrt[3]{4}\rho^{2}+\sqrt[3]{2}\rho

. … ►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})

. …

10: 21.5 Modular Transformations

… ►Let

\mathbf{A}

\mathbf{B}

\mathbf{C}

, and

\mathbf{D}

g\times g

matrices with integer elements such that ►

21.5.1

\boldsymbol{{\Gamma}}=\begin{bmatrix}\mathbf{A}&\mathbf{B}\\ \mathbf{C}&\mathbf{D}\end{bmatrix}

ⓘ

Defines:: Matrix $\boldsymbol{{\Gamma}}$ (locally)
Permalink:: http://dlmf.nist.gov/21.5.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §21.5(i), §21.5 and Ch.21

… ►Here

\xi(\boldsymbol{{\Gamma}})

is an eighth root of unity, that is,

(\xi(\boldsymbol{{\Gamma}}))^{8}=1

. … ►(

\mathbf{B}

symmetric with integer elements and even diagonal elements.) …(

\mathbf{B}

symmetric with integer elements.) …

.%E6%A2%85%E8%A5%BF%E5%92%8Cc%E7%BD%97%E7%AC%AC%E4%B8%80%E6%AC%A1%E4%B8%96%E7%95%8C%E6%9D%AF%E3%80%8E%E7%BD%91%E5%9D%80%3Amxsty.cc%E3%80%8F.%E6%97%B6%E4%B8%96%E7%95%8C%E6%9D%AF%E8%B5%9B%E7%A8%8B-m6q3s2-qmymcyumu.com

1—10 of 736 matching pages

1: 9.13 Generalized Airy Functions

2: Joyce E. Conlon

3: 19.32 Conformal Map onto a Rectangle

4: Bibliography Y

5: 19.37 Tables

Functions K ⁡ ( k ) and E ⁡ ( k )

Functions F ⁡ ( ϕ , k ) and E ⁡ ( ϕ , k )

6: Bibliography

7: 26.6 Other Lattice Path Numbers

Delannoy Number D ⁡ ( m , n )

8: 3.4 Differentiation

9: 1.11 Zeros of Polynomials

10: 21.5 Modular Transformations

Functions $K\left(k\right)$ and $E\left(k\right)$

Functions $F\left(\phi,k\right)$ and $E\left(\phi,k\right)$

Delannoy Number $D(m,n)$