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11—20 of 757 matching pages

11: 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

12: 3.4 Differentiation

… ►where

A_{k}^{n}

is as in (3.3.10). … ►If

f

can be extended analytically into the complex plane, then from Cauchy’s integral formula (§1.9(iii)) …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 …The integral on the right-hand side can be approximated by the composite trapezoidal rule (3.5.2). …

13: 9.4 Maclaurin Series

… ►

9.4.1

\operatorname{Ai}\left(z\right)=\operatorname{Ai}\left(0\right)\left(1+\frac{1% }{3!}z^{3}+\frac{1\cdot 4}{6!}z^{6}+\frac{1\cdot 4\cdot 7}{9!}z^{9}+\cdots% \right)+\operatorname{Ai}'\left(0\right)\left(z+\frac{2}{4!}z^{4}+\frac{2\cdot 5% }{7!}z^{7}+\frac{2\cdot 5\cdot 8}{10!}z^{10}+\cdots\right),

ⓘ

Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $!$ : factorial (as in $n!$ ) and $z$ : complex variable
Source:: Olver (1997b, p. 54)
A&S Ref:: 10.4.2 (with 10.4.4 and 10.4.5)
Permalink:: http://dlmf.nist.gov/9.4.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §9.4 and Ch.9

►

9.4.2

\operatorname{Ai}'\left(z\right)=\operatorname{Ai}'\left(0\right)\left(1+\frac% {2}{3!}z^{3}+\frac{2\cdot 5}{6!}z^{6}+\frac{2\cdot 5\cdot 8}{9!}z^{9}+\cdots% \right)+\operatorname{Ai}\left(0\right)\left(\frac{1}{2!}z^{2}+\frac{1\cdot 4}% {5!}z^{5}+\frac{1\cdot 4\cdot 7}{8!}z^{8}+\cdots\right),

ⓘ

Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $!$ : factorial (as in $n!$ ) and $z$ : complex variable
Proof sketch:: Use methods of Olver (1997b, p. 54).
Permalink:: http://dlmf.nist.gov/9.4.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §9.4 and Ch.9

►

9.4.3

\operatorname{Bi}\left(z\right)=\operatorname{Bi}\left(0\right)\left(1+\frac{1% }{3!}z^{3}+\frac{1\cdot 4}{6!}z^{6}+\frac{1\cdot 4\cdot 7}{9!}z^{9}+\cdots% \right)+\operatorname{Bi}'\left(0\right)\left(z+\frac{2}{4!}z^{4}+\frac{2\cdot 5% }{7!}z^{7}+\frac{2\cdot 5\cdot 8}{10!}z^{10}+\cdots\right),

ⓘ

Symbols:: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $!$ : factorial (as in $n!$ ) and $z$ : complex variable
Proof sketch:: Use methods of Olver (1997b, p. 54).
A&S Ref:: 10.4.3 (with 10.4.4 and 10.4.5)
Referenced by:: (9.12.15)
Permalink:: http://dlmf.nist.gov/9.4.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §9.4 and Ch.9

►

9.4.4

\operatorname{Bi}'\left(z\right)=\operatorname{Bi}'\left(0\right)\left(1+\frac% {2}{3!}z^{3}+\frac{2\cdot 5}{6!}z^{6}+\frac{2\cdot 5\cdot 8}{9!}z^{9}+\cdots% \right)+\operatorname{Bi}\left(0\right)\left(\frac{1}{2!}z^{2}+\frac{1\cdot 4}% {5!}z^{5}+\frac{1\cdot 4\cdot 7}{8!}z^{8}+\cdots\right).

ⓘ

Symbols:: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $!$ : factorial (as in $n!$ ) and $z$ : complex variable
Proof sketch:: Use methods of Olver (1997b, p. 54).
Permalink:: http://dlmf.nist.gov/9.4.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §9.4 and Ch.9

15: 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. …

Abramowitz and Stegun (1964, Chapter 8) tabulates $\mathsf{P}_{n}\left(x\right)$ for $n=0(1)3,9,10$ , $x=0(.01)1$ , 5–8D; $\mathsf{P}_{n}'\left(x\right)$ for $n=1(1)4,9,10$ , $x=0(.01)1$ , 5–7D; $\mathsf{Q}_{n}\left(x\right)$ and $\mathsf{Q}_{n}'\left(x\right)$ for $n=0(1)3,9,10$ , $x=0(.01)1$ , 6–8D; $P_{n}\left(x\right)$ and $P_{n}'\left(x\right)$ for $n=0(1)5,9,10$ , $x=1(.2)10$ , 6S; $Q_{n}\left(x\right)$ and $Q_{n}'\left(x\right)$ for $n=0(1)3,9,10$ , $x=1(.2)10$ , 6S. (Here primes denote derivatives with respect to $x$ .)

►

•

Zhang and Jin (1996, Chapter 4) tabulates $\mathsf{P}_{n}\left(x\right)$ for $n=2(1)5,10$ , $x=0(.1)1$ , 7D; $\mathsf{P}_{n}\left(\cos\theta\right)$ for $n=1(1)4,10$ , $\theta=0(5^{\circ})90^{\circ}$ , 8D; $\mathsf{Q}_{n}\left(x\right)$ for $n=0(1)2,10$ , $x=0(.1)0.9$ , 8S; $\mathsf{Q}_{n}\left(\cos\theta\right)$ for $n=0(1)3,10$ , $\theta=0(5^{\circ})90^{\circ}$ , 8D; $\mathsf{P}^{m}_{n}\left(x\right)$ for $m=1(1)4$ , $n-m=0(1)2$ , $n=10$ , $x=0,0.5$ , 8S; $\mathsf{Q}^{m}_{n}\left(x\right)$ for $m=1(1)4$ , $n=0(1)2,10$ , 8S; $\mathsf{P}^{m}_{\nu}\left(\cos\theta\right)$ for $m=0(1)3$ , $\nu=0(.25)5$ , $\theta=0(15^{\circ})90^{\circ}$ , 5D; $P_{n}\left(x\right)$ for $n=2(1)5,10$ , $x=1(1)10$ , 7S; $Q_{n}\left(x\right)$ for $n=0(1)2,10$ , $x=2(1)10$ , 8S. Corresponding values of the derivative of each function are also included, as are 6D values of the first 5 $\nu$ -zeros of $\mathsf{P}^{m}_{\nu}\left(\cos\theta\right)$ and of its derivative for $m=0(1)4$ , $\theta=10^{\circ},30^{\circ},150^{\circ}$ .

►

•

Belousov (1962) tabulates $\mathsf{P}^{m}_{n}\left(\cos\theta\right)$ (normalized) for $m=0(1)36$ , $n-m=0(1)56$ , $\theta=0(2.5^{\circ})90^{\circ}$ , 6D.

…

17: 34 3j, 6j, 9j Symbols

Chapter 34 $\mathit{3j},\mathit{6j},\mathit{9j}$ Symbols

…

18: 8 Incomplete Gamma and Related
Functions

Chapter 8 Incomplete Gamma and Related Functions

…

19: 26.5 Lattice Paths: Catalan Numbers

… ►

C\left(n\right)

is the Catalan number. …(Sixty-six equivalent definitions of

C\left(n\right)

are given in Stanley (1999, pp. 219–229).) … ►

26.5.3

C\left(n+1\right)=\sum_{k=0}^{n}C\left(k\right)C\left(n-k\right),

ⓘ

Symbols:: $C\left(\NVar{n}\right)$ : Catalan number, $k$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.5.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §26.5(iii), §26.5 and Ch.26

►

26.5.4

C\left(n+1\right)=\frac{2(2n+1)}{n+2}C\left(n\right),

ⓘ

Symbols:: $C\left(\NVar{n}\right)$ : Catalan number and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.5.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §26.5(iii), §26.5 and Ch.26

… ►

26.5.7

\lim_{n\to\infty}\frac{C\left(n+1\right)}{C\left(n\right)}=4.

ⓘ

Symbols:: $C\left(\NVar{n}\right)$ : Catalan number and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.5.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §26.5(iv), §26.5 and Ch.26

20: 12.14 The Function $W\left(a,x\right)$

… ►For the modulus functions

\widetilde{F}(a,x)

and

\widetilde{G}(a,x)

see §12.14(x). … ►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). …

11—20 of 757 matching pages

11: 1.11 Zeros of Polynomials

12: 3.4 Differentiation

13: 9.4 Maclaurin Series

14: 3.3 Interpolation

15: 26.10 Integer Partitions: Other Restrictions

16: 14.33 Tables

17: 34 3j, 6j, 9j Symbols

Chapter 34 $\mathit{3j},\mathit{6j},\mathit{9j}$ Symbols

18: 8 Incomplete Gamma and Related
Functions

Chapter 8 Incomplete Gamma and Related Functions

19: 26.5 Lattice Paths: Catalan Numbers

20: 12.14 The Function $W\left(a,x\right)$

11—20 of 757 matching pages

11: 1.11 Zeros of Polynomials

12: 3.4 Differentiation

13: 9.4 Maclaurin Series

14: 3.3 Interpolation

15: 26.10 Integer Partitions: Other Restrictions

16: 14.33 Tables

17: 34 3j, 6j, 9j Symbols

Chapter 34 3 ⁢ j , 6 ⁢ j , 9 ⁢ j Symbols

18: 8 Incomplete Gamma and Related Functions

Chapter 8 Incomplete Gamma and Related Functions

19: 26.5 Lattice Paths: Catalan Numbers

20: 12.14 The Function W ⁡ ( a , x )

Chapter 34 $\mathit{3j},\mathit{6j},\mathit{9j}$ Symbols

18: 8 Incomplete Gamma and Related
Functions

20: 12.14 The Function $W\left(a,x\right)$