2022年福彩3D历史上的今天开奖号085期【杏彩官方qee9.com】yGfq2Nne

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21: 1.11 Zeros of Polynomials

… ►

${\displaystyle \frac{-b\pm \sqrt{D}}{2a}},$

►

$D=b^2-4ac.$

… ►Let ►

$D_1=a_1,$

… ►Then $f(z)$, with $a_n\ne 0$, is stable iff $a_0\ne 0$; $D_{2k}>0$, $k=1,\mathrm{\dots },\lfloor \frac{1}{2}n\rfloor$; $\mathrm{sign}{D}_{2k+1}=\mathrm{sign}{a}_0$, $k=0,1,\mathrm{\dots },\lfloor \frac{1}{2}n-\frac{1}{2}\rfloor$.

22: 21.5 Modular Transformations

… ►Let $𝐀$, $𝐁$, $𝐂$, and $𝐃$ be $g\times g$ matrices with integer elements such that ►

21.5.1

ⓘ

Defines:

Matrix $𝚪$ (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

… ►

21.5.4 $$\theta \left({\left[{[𝐂𝛀+𝐃]}^{-1}\right]}^{\mathrm{T}}𝐳\right|[𝐀𝛀+𝐁]{[𝐂𝛀+𝐃]}^{-1})=\xi (𝚪)\sqrt{det[𝐂𝛀+𝐃]}{\mathrm{e}}^{\pi \mathrm{i}𝐳\cdot \left[{[𝐂𝛀+𝐃]}^{-1}𝐂\right]\cdot 𝐳}\theta \left(𝐳\right|𝛀).$$

ⓘ

Symbols:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $det$: determinant, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${𝐀}^{\mathrm{T}}$: transpose of matrix, $𝛀$: a Riemann matrix, Matrix $𝚪$ and $\xi (𝚪)$: eighth root of unity

Referenced by:

§21.5(i)

Permalink:

http://dlmf.nist.gov/21.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §21.5(i), §21.5 and Ch.21

… ►

21.5.9 $$\theta \left[\genfrac{}{}{0.0pt}{}{𝐃𝜶-𝐂𝜷+\frac{1}{2}\mathrm{diag}[𝐂{𝐃}^{\mathrm{T}}]}{-𝐁𝜶+𝐀𝜷+\frac{1}{2}\mathrm{diag}[𝐀{𝐁}^{\mathrm{T}}]}\right]\left({\left[{[𝐂𝛀+𝐃]}^{-1}\right]}^{\mathrm{T}}𝐳\right|[𝐀𝛀+𝐁]{[𝐂𝛀+𝐃]}^{-1})=\kappa (𝜶,𝜷,𝚪)\sqrt{det[𝐂𝛀+𝐃]}{\mathrm{e}}^{\pi \mathrm{i}𝐳\cdot \left[{[𝐂𝛀+𝐃]}^{-1}𝐂\right]\cdot 𝐳}\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀),$$

ⓘ

Symbols:

$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics, $\pi$: the ratio of the circumference of a circle to its diameter, $det$: determinant, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${𝐀}^{\mathrm{T}}$: transpose of matrix, $𝛀$: a Riemann matrix, $𝜶$: $g$-dimensional vector, $𝜷$: $g$-dimensional vector, $\mathrm{diag}𝐀$: Transpose of $[A_{11},A_{22},\mathrm{\dots },A_{gg}]$ and Matrix $𝚪$

Permalink:

http://dlmf.nist.gov/21.5.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §21.5(ii), §21.5 and Ch.21

…

23: 29.6 Fourier Series

… ►

29.6.39 $$({\beta }_0-H)D_1+{\alpha }_0{D}_3=0,$$

ⓘ

Symbols:

${\alpha }_p$ , ${\beta }_p$ , $D_{2p+1}$: coefficents and $H$

Permalink:

http://dlmf.nist.gov/29.6.E39

Encodings:

pMML, png, TeX

See also:

Annotations for §29.6(iii), §29.6 and Ch.29

… ►

29.6.43 $$\sum _{p=0}^{\mathrm{\infty }}(2p+1)D_{2p+1}>0,$$

ⓘ

Symbols:

$p$: nonnegative integer and $D_{2p+1}$: coefficents

Permalink:

http://dlmf.nist.gov/29.6.E43

Encodings:

pMML, png, TeX

See also:

Annotations for §29.6(iii), §29.6 and Ch.29

… ►

29.6.54 $$({\beta }_0-H)D_2+{\alpha }_0{D}_4=0,$$

ⓘ

Symbols:

${\alpha }_p$ , ${\beta }_p$ , $H$ and $D_{2p}$: coefficents

Permalink:

http://dlmf.nist.gov/29.6.E54

Encodings:

pMML, png, TeX

See also:

Annotations for §29.6(iv), §29.6 and Ch.29

… ►

29.6.57 $$\left(1-\frac{1}{2}{k}^2\right)\sum _{p=1}^{\mathrm{\infty }}{D}_{2p}^2-\frac{1}{2}{k}^2\sum _{p=1}^{\mathrm{\infty }}{D}_{2p}{D}_{2p+2}=1,$$

ⓘ

Symbols:

$p$: nonnegative integer, $k$: real parameter and $D_{2p}$: coefficents

Permalink:

http://dlmf.nist.gov/29.6.E57

Encodings:

pMML, png, TeX

See also:

Annotations for §29.6(iv), §29.6 and Ch.29

►

29.6.58 $$\sum _{p=0}^{\mathrm{\infty }}(2p+2)D_{2p+2}>0,$$

ⓘ

Symbols:

$p$: nonnegative integer and $D_{2p}$: coefficents

Permalink:

http://dlmf.nist.gov/29.6.E58

Encodings:

pMML, png, TeX

See also:

Annotations for §29.6(iv), §29.6 and Ch.29

…

24: 29.15 Fourier Series and Chebyshev Series

… ►

29.15.34 $${[D_1,D_3,\mathrm{\dots },D_{2n+1}]}^{\mathrm{T}},$$

ⓘ

Symbols:

$n$: nonnegative integer and $D_{2p}$: coefficents

Permalink:

http://dlmf.nist.gov/29.15.E34

Encodings:

pMML, png, TeX

See also:

Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

►

29.15.35 $$\left(1-\frac{1}{2}{k}^2\right)\sum _{p=0}^n{D}_{2p+1}^2+\frac{1}{2}{k}^2\left(\frac{1}{2}{D}_1^2-\sum _{p=0}^{n-1}{D}_{2p+1}{D}_{2p+3}\right)=1,$$

ⓘ

Symbols:

$n$: nonnegative integer, $p$: nonnegative integer, $k$: real parameter and $D_{2p}$: coefficents

Permalink:

http://dlmf.nist.gov/29.15.E35

Encodings:

pMML, png, TeX

See also:

Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

►

29.15.36 $$\sum _{p=0}^n(2p+1)D_{2p+1}>0.$$

ⓘ

Symbols:

$n$: nonnegative integer, $p$: nonnegative integer and $D_{2p}$: coefficents

Permalink:

http://dlmf.nist.gov/29.15.E36

Encodings:

pMML, png, TeX

See also:

Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

… ►

29.15.39 $${[D_2,D_4,\mathrm{\dots },D_{2n+2}]}^{\mathrm{T}},$$

ⓘ

Symbols:

$n$: nonnegative integer and $D_{2p}$: coefficents

Permalink:

http://dlmf.nist.gov/29.15.E39

Encodings:

pMML, png, TeX

See also:

Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

… ►

29.15.41 $$\sum _{p=0}^n(2p+2)D_{2p+2}>0.$$

ⓘ

Symbols:

$n$: nonnegative integer, $p$: nonnegative integer and $D_{2p}$: coefficents

Permalink:

http://dlmf.nist.gov/29.15.E41

Encodings:

pMML, png, TeX

See also:

Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

…

25: 10.13 Other Differential Equations

… ►In (10.13.9)–(10.13.11) ${𝒞}_{\nu }\left(z\right)$, ${𝒟}_{\mu }(z)$ are any cylinder functions of orders $\nu ,\mu$, respectively, and $\mathit{\vartheta }=z(d/dz)$. ►

10.13.9 $$z^2{w}^{\prime \prime \prime }+3z{w}^{\prime \prime }+(4z^2+1-4{\nu }^2)w^{\prime }+4zw=0,$$ $w={𝒞}_{\nu }\left(z\right){𝒟}_{\nu }(z)$,

ⓘ

Symbols:

${𝒞}_{\nu }\left(z\right)$: cylinder function, $z$: complex variable, $\nu$: complex parameter and ${𝒟}_{\nu }(z)$: cylinder function

A&S Ref:

9.1.58 (modified)

Referenced by:

§10.13, §10.36

Permalink:

http://dlmf.nist.gov/10.13.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §10.13 and Ch.10

►

10.13.10 $$z^3{w}^{\prime \prime \prime }+z(4z^2+1-4{\nu }^2)w^{\prime }+(4{\nu }^2-1)w=0,$$ $w=z{𝒞}_{\nu }\left(z\right){𝒟}_{\nu }(z)$,

ⓘ

Symbols:

${𝒞}_{\nu }\left(z\right)$: cylinder function, $z$: complex variable, $\nu$: complex parameter and ${𝒟}_{\nu }(z)$: cylinder function

A&S Ref:

9.1.59

Permalink:

http://dlmf.nist.gov/10.13.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §10.13 and Ch.10

►

10.13.11 $$\left({\mathit{\vartheta }}^4-2({\nu }^2+{\mu }^2){\mathit{\vartheta }}^2+{({\nu }^2-{\mu }^2)}^2\right)w+4z^2(\mathit{\vartheta }+1)(\mathit{\vartheta }+2)w=0,$$ $w={𝒞}_{\nu }\left(z\right){𝒟}_{\mu }(z).$

ⓘ

Symbols:

${𝒞}_{\nu }\left(z\right)$: cylinder function, $z$: complex variable, $\nu$: complex parameter, $\mathit{\vartheta }=z(d/dz)$ and ${𝒟}_{\nu }(z)$: cylinder function

A&S Ref:

9.1.57

Referenced by:

§10.13, §10.36

Permalink:

http://dlmf.nist.gov/10.13.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §10.13 and Ch.10

…

26: 19.4 Derivatives and Differential Equations

… ►Let $D_k=\partial /\partial k$. Then ►

19.4.8 $$(kk_{}^{\prime }{}_{}{}^2{D}_k^2+(1-3k^2)D_k-k)F(\varphi ,k)=\frac{-k\sin \varphi \cos \varphi }{{(1-k^2{\sin }^2\varphi )}^{3/2}},$$

ⓘ

Symbols:

$\cos z$: cosine function, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus, $k^{\prime }$: complementary modulus and $D_k$: differential operator

Referenced by:

§19.4(ii)

Permalink:

http://dlmf.nist.gov/19.4.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.4(ii), §19.4 and Ch.19

►

19.4.9 $$(kk_{}^{\prime }{}_{}{}^2{D}_k^2+k_{}^{\prime }{}_{}{}^2{D}_k+k)E(\varphi ,k)=\frac{k\sin \varphi \cos \varphi }{\sqrt{1-k^2{\sin }^2\varphi }}.$$

ⓘ

Symbols:

$\cos z$: cosine function, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus, $k^{\prime }$: complementary modulus and $D_k$: differential operator

Permalink:

http://dlmf.nist.gov/19.4.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §19.4(ii), §19.4 and Ch.19

…

27: 26.21 Tables

… ►Abramowitz and Stegun (1964, Chapter 24) tabulates binomial coefficients $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$ for $m$ up to 50 and $n$ up to 25; extends Table 26.4.1 to $n=10$; tabulates Stirling numbers of the first and second kinds, $s(n,k)$ and $S(n,k)$, for $n$ up to 25 and $k$ up to $n$; tabulates partitions $p\left(n\right)$ and partitions into distinct parts $p(𝒟,n)$ for $n$ up to 500. …

28: 28.8 Asymptotic Expansions for Large $q$

… ►Also let $\xi =2\sqrt{h}\cos x$ and $D_m\left(\xi \right)={\mathrm{e}}^{-{\xi }^2/4}{\mathrm{𝐻𝑒}}_m\left(\xi \right)$ (§18.3). … ►

28.8.4 $$U_m(\xi )\sim D_m\left(\xi \right)-\frac{1}{2^{6}h}\left(D_{m+4}\left(\xi \right)-4!\left(\genfrac{}{}{0pt}{}{m}{4}\right)D_{m-4}\left(\xi \right)\right)+\frac{1}{2^{13}{h}^2}\left(D_{m+8}\left(\xi \right)-2^5(m+2)D_{m+4}\left(\xi \right)+4!{\mathrm{\hspace{0.17em}2}}^5(m-1)\left(\genfrac{}{}{0pt}{}{m}{4}\right)D_{m-4}\left(\xi \right)+8!\left(\genfrac{}{}{0.0pt}{}{m}{8}\right)D_{m-8}\left(\xi \right)\right)+\mathrm{\cdots },$$

ⓘ

Defines:

$U_m(\xi )$: function (locally)

Symbols:

$D_{\nu }\left(z\right)$: parabolic cylinder function, $\sim$: Poincaré asymptotic expansion, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $!$: factorial (as in $n!$), $m$: integer, $h$: parameter and $\xi$: variable

A&S Ref:

20.9.17 (in different form)

Permalink:

http://dlmf.nist.gov/28.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.8(ii), §28.8 and Ch.28

►

28.8.5 $$V_m(\xi )\sim \frac{1}{2^{4}h}\left(-D_{m+2}\left(\xi \right)-m(m-1)D_{m-2}\left(\xi \right)\right)+\frac{1}{2^{10}{h}^2}\left(D_{m+6}\left(\xi \right)+(m^2-25m-36)D_{m+2}\left(\xi \right)-m(m-1)(m^2+27m-10)D_{m-2}\left(\xi \right)-6!\left(\genfrac{}{}{0.0pt}{}{m}{6}\right)D_{m-6}\left(\xi \right)\right)+\mathrm{\cdots },$$

ⓘ

Defines:

$V_m(\xi )$: function (locally)

Symbols:

$D_{\nu }\left(z\right)$: parabolic cylinder function, $\sim$: Poincaré asymptotic expansion, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $!$: factorial (as in $n!$), $m$: integer, $h$: parameter and $\xi$: variable

A&S Ref:

20.9.18 (in slightly different form)

Referenced by:

Erratum (V1.0.16) for Equation (28.8.5)

Permalink:

http://dlmf.nist.gov/28.8.E5

Encodings:

pMML, png, TeX

Error (effective with 1.0.16):

Originally the $-$ in front of the $6!$ was given incorrectly as $+$.

Suggested 2017-02-02 by Daniel Karlsson

See also:

Annotations for §28.8(ii), §28.8 and Ch.28

…

29: 1.5 Calculus of Two or More Variables

… ►A function is continuous on a point set $D$ if it is continuous at all points of $D$. … ►If $f(x,y)$ is continuous, and $D$ is the set … ►Similarly, if $D$ is the set … ►If $D$ can be represented in both forms (1.5.30) and (1.5.33), and $f(x,y)$ is continuous on $D$, then … ►Infinite double integrals occur when $f(x,y)$ becomes infinite at points in $D$ or when $D$ is unbounded. …

30: 4.43 Cubic Equations

… ►

4.43.2 $$z^3+pz+q=0$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $a$: real or complex constant, $z$: complex variable, $p$: real constant and $q$: real constant

Referenced by:

(4.43.1), §4.43

Permalink:

http://dlmf.nist.gov/4.43.E2

Encodings:

pMML, png, TeX

Errata (effective with 1.0.10):

Cases (a), (b), and (c) of of this equation have been corrected. Formerly, with constants $C$ and $D$ given in Eq. (4.43.1), they read

1. $A\sin a$, $A\sin \left(a+\frac{2}{3}\pi \right)$, and $A\sin \left(a+\frac{4}{3}\pi \right)$, with $\sin \left(3a\right)=C$, when and $C\le 1$.

2. $A\cosh a$, $A\cosh \left(a+\frac{2}{3}\pi \mathrm{i}\right)$, and $A\cosh \left(a+\frac{4}{3}\pi \mathrm{i}\right)$, with $\cosh \left(3a\right)=C$, when and $C>1$.

3. $B\sinh a$, $B\sinh \left(a+\frac{2}{3}\pi \mathrm{i}\right)$, and $B\sinh \left(a+\frac{4}{3}\pi \mathrm{i}\right)$, with $\sinh \left(3a\right)=D$, when $p>0$.

Reported 2014-10-31 by Masataka Urago

See also:

Annotations for §4.43 and Ch.4

…