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21: 1.11 Zeros of Polynomials

… ►A monic polynomial of even degree with real coefficients has at least two zeros of opposite signs when the constant term is negative. … ►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})

. … ►with real coefficients, is called stable if the real parts of all the zeros are strictly negative. …

22: 32.8 Rational Solutions

… ►

32.8.3

w(z;3)=\frac{3z^{2}}{z^{3}+4}-\frac{6z^{2}(z^{3}+10)}{z^{6}+20z^{3}-80},

ⓘ

Symbols:: $z$ : real
Permalink:: http://dlmf.nist.gov/32.8.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §32.8(ii), §32.8 and Ch.32

►

32.8.4

w(z;4)=-\frac{1}{z}+\frac{6z^{2}(z^{3}+10)}{z^{6}+20z^{3}-80}-\frac{9z^{5}(z^{% 3}+40)}{z^{9}+60z^{6}+11200}.

ⓘ

Symbols:: $z$ : real
Permalink:: http://dlmf.nist.gov/32.8.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §32.8(ii), §32.8 and Ch.32

… ►

Q_{3}(z)=z^{6}+20z^{3}-80,

… ►

(c)

$\alpha=\tfrac{1}{2}a^{2}$ , $\beta=-\tfrac{1}{2}(a+n)^{2}$ , and $\gamma=m$ , with $m+n$ even.

►

(d)

$\alpha=\tfrac{1}{2}(b+n)^{2}$ , $\beta=-\tfrac{1}{2}b^{2}$ , and $\gamma=m$ , with $m+n$ even.

…

23: 19.36 Methods of Computation

… ►

19.36.2

1-\tfrac{3}{14}E_{2}+\tfrac{1}{6}E_{3}+\tfrac{9}{88}E_{2}^{2}-\tfrac{3}{22}E_{% 4}-\tfrac{9}{52}E_{2}E_{3}+\tfrac{3}{26}E_{5}-\tfrac{1}{16}E_{2}^{3}+\tfrac{3}% {40}E_{3}^{2}+\tfrac{3}{20}E_{2}E_{4}+\tfrac{45}{272}E_{2}^{2}E_{3}-\tfrac{9}{% 68}(E_{3}E_{4}+E_{2}E_{5}).

ⓘ

Symbols:: $E_{s}(\mathbf{z})$ : symmetric function
Referenced by:: §19.19
Permalink:: http://dlmf.nist.gov/19.36.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §19.36(i), §19.36 and Ch.19

… ►Because

U_{12}

may be real and negative, or even complex, care is needed to ensure

\sqrt{x}=U_{12}

, and similarly for

y

and

z

. … ►For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20). … ►A three-part computational procedure for

\Pi\left(\phi,\alpha^{2},k\right)

is described by Franke (1965) for

\alpha^{2}<1

. …

24: 26.9 Integer Partitions: Restricted Number and Part Size

§26.9 Integer Partitions: Restricted Number and Part Size

… ►

p_{k}\left(n\right)

denotes the number of partitions of

n

into at most

k

parts. … … ►Conjugation establishes a one-to-one correspondence between partitions of

n

into at most

k

parts and partitions of

n

into parts with largest part less than or equal to

k

. … ►equivalently, partitions into at most

k

parts either have exactly

k

parts, in which case we can subtract one from each part, or they have strictly fewer than

k

parts. …

25: 36.8 Convergent Series Expansions

… ►

\Psi_{K}\left(\mathbf{x}\right)=\dfrac{2}{K+2}\sum\limits_{n=0}^{\infty}\exp% \left(i\dfrac{\pi(2n+1)}{2(K+2)}\right)\Gamma\left(\dfrac{2n+1}{K+2}\right)a_{% 2n}(\mathbf{x}),

K

even,

… ►

36.8.4

\Psi^{(\mathrm{E})}\left(\mathbf{x}\right)=2\pi^{2}\left(\dfrac{2}{3}\right)^{% 2/3}\sum\limits_{n=0}^{\infty}\dfrac{\left(-i(2/3)^{2/3}z\right)^{n}}{n!}\Re% \left(f_{n}\left(\dfrac{x+iy}{12^{1/3}},\dfrac{x-iy}{12^{1/3}}\right)\right),

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\Psi^{(\mathrm{E})}\left(\NVar{\mathbf{x}}\right)$ : elliptic umbilic canonical integral function, $!$ : factorial (as in $n!$ ), $\mathrm{i}$ : imaginary unit, $\Re$ : real part, $y$ : real parameter, $z$ : real parameter, $m$ : integer, $x$ : real parameter and $f_{n}$ : function
Referenced by:: §36.8
Permalink:: http://dlmf.nist.gov/36.8.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §36.8 and Ch.36

…

27: 20.11 Generalizations and Analogs

… ►For relatively prime integers

m, n

with

n>0

and

m n

even, the Gauss sum

G(m,n)

is defined by …

28: 30.9 Asymptotic Approximations and Expansions

… ►

2^{20}\beta_{5}=-527q^{7}-61529q^{5}-10\;43961q^{3}-22\;41599q+32m^{2}(5739q^{% 5}+1\;27550q^{3}+2\;98951q)-2048m^{4}(355q^{3}+1505q)+65536m^{6}q.

… ►As

\gamma^{2}\to-\infty

, with

q=n+1

n-m

is even, or

q=n

n-m

is odd, we have …

29: 23.18 Modular Transformations

… ►Here e and o are generic symbols for even and odd integers, respectively. In particular, if

a-1,b,c

, and

d-1

are all even, then …

Zhang and Jin (1996, p. 337) tabulates $\operatorname{Ai}\left(x\right)$ , $\operatorname{Ai}'\left(x\right)$ , $\operatorname{Bi}\left(x\right)$ , $\operatorname{Bi}'\left(x\right)$ for $x=0(1)20$ to 8S and for $x=-20(1)0$ to 9D.

… ►

•

Woodward and Woodward (1946) tabulates the real and imaginary parts of $\operatorname{Ai}\left(z\right)$ , $\operatorname{Ai}'\left(z\right)$ , $\operatorname{Bi}\left(z\right)$ , $\operatorname{Bi}'\left(z\right)$ for $\Re z=-2.4(.2)2.4$ , $\Im z=-2.4(.2)0$ . Precision is 4D.

… ►

•

Miller (1946) tabulates $a_{k}$ , $\operatorname{Ai}'\left(a_{k}\right)$ , $a^{\prime}_{k}$ , $\operatorname{Ai}\left(a^{\prime}_{k}\right)$ , $k=1(1)50$ ; $b_{k}$ , $\operatorname{Bi}'\left(b_{k}\right)$ , $b^{\prime}_{k}$ , $\operatorname{Bi}\left(b^{\prime}_{k}\right)$ , $k=1(1)20$ . Precision is 8D. Entries for $k=1(1)20$ are reproduced in Abramowitz and Stegun (1964, Chapter 10).

►

•

Sherry (1959) tabulates $a_{k}$ , $\operatorname{Ai}'\left(a_{k}\right)$ , $a^{\prime}_{k}$ , $\operatorname{Ai}\left(a^{\prime}_{k}\right)$ , $k=1(1)50$ ; 20S.

… ►

•

Corless et al. (1992) gives the real and imaginary parts of $\beta_{k}$ for $k=1(1)13$ ; 14S.

…