zelle support number +1(888‒858‒1410)

SearchAdvancedHelp

Did you mean zelene support number +1(888‒858‒1410) ?

(0.011 seconds)

1—10 of 83 matching pages

1: 31.14 General Fuchsian Equation

… ►The general second-order Fuchsian equation with $N+1$ regular singularities at $z=a_j$, $j=1,2,\mathrm{\dots },N$, and at $\mathrm{\infty }$, is given by ►

31.14.1 $$\frac{d^{2}w}{{dz}^2}+\left(\sum _{j=1}^N\frac{{\gamma }_j}{z-a_j}\right)\frac{dw}{dz}+\left(\sum _{j=1}^N\frac{q_j}{z-a_j}\right)w=0,$$ ${\sum }_{j=1}^N{q}_j=0$.

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter and $N+1$: number of singularities

Referenced by:

§31.15(i)

Permalink:

http://dlmf.nist.gov/31.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.14(i), §31.14 and Ch.31

… ►

31.14.3 $$w(z)=\left(\prod _{j=1}^N{(z-a_j)}^{-{\gamma }_j/2}\right)W(z),$$

ⓘ

Symbols:

$z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $a$: complex parameter, $N+1$: number of singularities and $W(z)$: solution

Permalink:

http://dlmf.nist.gov/31.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §31.14(i), §31.14(i), §31.14 and Ch.31

►

31.14.4 $$\frac{d^{2}W}{{dz}^2}=\sum _{j=1}^N\left(\frac{{\tilde{\gamma }}_j}{{(z-a_j)}^2}+\frac{{\tilde{q}}_j}{z-a_j}\right)W,$$ ${\sum }_{j=1}^N{\tilde{q}}_j=0$,

ⓘ

Defines:

$W(z)$: solution (locally)

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter and $N+1$: number of singularities

Permalink:

http://dlmf.nist.gov/31.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §31.14(i), §31.14(i), §31.14 and Ch.31

…

2: Customize DLMF

…Note: Javascript and Cookies must be enabled to support customizations.

3: 31.15 Stieltjes Polynomials

… ►

31.15.2 $$\sum _{j=1}^N\frac{{\gamma }_j/2}{z_k-a_j}+\sum _{\begin{array}{c}j=1\\ j\ne k\end{array}}^n\frac{1}{z_k-z_j}=0,$$ $k=1,2,\mathrm{\dots },n$.

ⓘ

Symbols:

$z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter and $N+1$: number of singularities

Referenced by:

§31.15(ii)

Permalink:

http://dlmf.nist.gov/31.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.15(ii), §31.15 and Ch.31

… ►

31.15.3 $$\sum _{j=1}^N\frac{{\gamma }_j}{t_k-a_j}+\sum _{j=1}^{n-1}\frac{1}{t_k-z_j^{\prime }}=0.$$

ⓘ

Symbols:

$z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $N+1$: number of singularities and $t_k$: zero of $V(z)$

Permalink:

http://dlmf.nist.gov/31.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §31.15(ii), §31.15 and Ch.31

… ►

31.15.6 $j=1,2,\mathrm{\dots },N-1$,

ⓘ

Symbols:

$j$: nonnegative integer, $a$: complex parameter and $N+1$: number of singularities

Referenced by:

§31.15(iii)

Permalink:

http://dlmf.nist.gov/31.15.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §31.15(ii), §31.15 and Ch.31

… ►

31.15.7 $$q_j={\gamma }_j\sum _{k=1}^n\frac{1}{z_k-a_j},$$ $j=1,2,\mathrm{\dots },N$.

ⓘ

Symbols:

$z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter and $N+1$: number of singularities

Permalink:

http://dlmf.nist.gov/31.15.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §31.15(ii), §31.15 and Ch.31

… ►

31.15.8 $$S_{𝐦}(z_1)S_{𝐦}(z_2)\mathrm{\cdots }{S}_{𝐦}(z_{N-1}),$$ $z_j\in (a_j,a_{j+1})$,

ⓘ

Symbols:

$\in$: element of, $(a,b)$: open interval, $z$: complex variable, $j$: nonnegative integer, $a$: complex parameter, $N+1$: number of singularities and $S_{𝐦}(z)$: Stieltjes polynomial

Referenced by:

§31.15(iii)

Permalink:

http://dlmf.nist.gov/31.15.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §31.15(iii), §31.15 and Ch.31

…

4: About MathML

… ►, built-in to the browser) support for MathML is growing, (see Browsers supporting MathML). … ►

Browsers supporting MathML

►The Firefox browser has traditionally had the strongest support for MathML and its native MathML is used by default. ►Recent enhancements to the WebKit engine now provide support for MathML Core. … ►Most modern browsers support ‘Web Fonts’, fonts that are effectively included with a web site. …

5: 26.17 The Twelvefold Way

… ►

Table 26.17.1: The twelvefold way.

► ►►►►

elements of $N$	elements of $K$	$f$ unrestricted	$f$ one-to-one	$f$ onto
labeled	labeled	$k^n$	${\left(k-n+1\right)}_n$	$k!S(n,k)$
…
labeled	unlabeled	$\begin{array}{c}S(n,1)+S(n,2)\\ +\mathrm{\cdots }+S(n,k)\end{array}$		$S(n,k)$
…

►

6: 26.11 Integer Partitions: Compositions

… ►For example, there are eight compositions of 4: $4,3+1,1+3,2+2,2+1+1,1+2+1,1+1+2$, and $1+1+1+1$. $c\left(n\right)$ denotes the number of compositions of $n$, and $c_m\left(n\right)$ is the number of compositions into exactly $m$ parts. $c(\in T,n)$ is the number of compositions of $n$ with no 1’s, where again $T=\{2,3,4,\mathrm{\dots }\}$. … ►The Fibonacci numbers are determined recursively by … ►Additional information on Fibonacci numbers can be found in Rosen et al. (2000, pp. 140–145).

7: 26.5 Lattice Paths: Catalan Numbers

… ►

26.5.1 $$C\left(n\right)=\frac{1}{n+1}\left(\genfrac{}{}{0.0pt}{}{2n}{n}\right)=\frac{1}{2n+1}\left(\genfrac{}{}{0.0pt}{}{2n+1}{n}\right)=\left(\genfrac{}{}{0.0pt}{}{2n}{n}\right)-\left(\genfrac{}{}{0.0pt}{}{2n}{n-1}\right)=\left(\genfrac{}{}{0.0pt}{}{2n-1}{n}\right)-\left(\genfrac{}{}{0.0pt}{}{2n-1}{n+1}\right).$$

ⓘ

Defines:

$C\left(n\right)$: Catalan number

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient and $n$: nonnegative integer

Referenced by:

§26.5(iv)

Permalink:

http://dlmf.nist.gov/26.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §26.5(i), §26.5 and Ch.26

… ►

26.5.3 $$C\left(n+1\right)=\sum _{k=0}^{n}C\left(k\right)C\left(n-k\right),$$

ⓘ

Symbols:

$C\left(n\right)$: 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:

$C\left(n\right)$: 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

►

26.5.5 $$C\left(n+1\right)=\sum _{k=0}^{\lfloor n/2\rfloor }\left(\genfrac{}{}{0.0pt}{}{n}{2k}\right)2^{n-2k}C\left(k\right).$$

ⓘ

Symbols:

$C\left(n\right)$: Catalan number, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\lfloor x\rfloor$: floor of $x$ , $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §26.5(iii), §26.5 and Ch.26

… ►

26.5.7 $$\underset{n\to \mathrm{\infty }}{lim}\frac{C\left(n+1\right)}{C\left(n\right)}=4.$$

ⓘ

Symbols:

$C\left(n\right)$: 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

8: 27.18 Methods of Computation: Primes

… ►These algorithms are used for testing primality of Mersenne numbers, $2^n-1$, and Fermat numbers, $2^{2^n}+1$. …

9: Viewing DLMF Interactive 3D Graphics

… ►WebGL is supported in the current versions of most common web browsers. … ►1, some advanced features of X3DOM are currently not fully supported (see x3dom.org). …If you have trouble viewing the WebGL visualizations in your web browser, see x3dom.org or caniuse.com/webgl for information on WebGL browser support. … ►Please see caniuse.com/webgl or x3dom.org for information on WebGL browser support.

10: 35 Functions of Matrix Argument

…