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1: Viewing DLMF Interactive 3D Graphics

… ►Users can render a 3D scene and interactively rotate, scale, and otherwise explore a function surface. … ►WebGL is a JavaScript API (application programming interface) for rendering 3D graphics in a web browser without the use of a plugin. … ►If you prefer to use a web browser that is not WebGL-enabled, we currently also provide VRML and X3D versions of the DLMF visualizations that may be viewed with a special VRML/X3D browser or plugin. …Any installed VRML or X3D browser should be removed before installing a new one. … ►Linux users must view the visualizations in a WebGL-enabled browser. …

2: 27.15 Chinese Remainder Theorem

… ►The Chinese remainder theorem states that a system of congruences $x\equiv a_1(mod{m}_1),\mathrm{\dots },x\equiv a_k(mod{m}_k)$, always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod $m$), where $m$ is the product of the moduli. … ►For example, suppose a lengthy calculation involves many 10-digit integers. …Because each residue has no more than five digits, the arithmetic can be performed efficiently on these residues with respect to each of the moduli, yielding answers $a_1(mod{m}_1)$, $a_2(mod{m}_2)$, $a_3(mod{m}_3)$, and $a_4(mod{m}_4)$, where each $a_j$ has no more than five digits. … ►Even though the lengthy calculation is repeated four times, once for each modulus, most of it only uses five-digit integers and is accomplished quickly without overwhelming the machine’s memory. Details of a machine program describing the method together with typical numerical results can be found in Newman (1967). …

3: Foreword

… ►The provision of standard reference data of this type is a core function of NIST. … ►The new printed volume, the NIST Handbook of Mathematical Functions, serves a similar function as the original A&S, though it is heavily updated and extended. …The DLMF may well serve as a model for the effective presentation of highly mathematical reference material on the Web. ►The production of these new resources has been a very complex undertaking some 10 years in the making. This could not have been done without the cooperation of many mathematicians, information technologists, and physical scientists both within NIST and externally. …

4: 15.13 Zeros

… ►Let $N(a,b,c)$ denote the number of zeros of $F(a,b;c;z)$ in the sector . If $a$, $b$, $c$ are real, $a$, $b$, $c$, $c-a$, $c-b\ne 0,-1,-2,\mathrm{\dots }$, and, without loss of generality, $b\ge a$, $c\ge a+b$ (compare (15.8.1)), then … ►where $S=\mathrm{sign}\left(\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(c-a\right)\mathrm{\Gamma }\left(c-b\right)\right)$. … ►If $a$, $b$, $c$, $c-a$, or $c-b\in \{0,-1,-2,\mathrm{\dots }\}$, then $F(a,b;c;z)$ is not defined, or reduces to a polynomial, or reduces to ${(1-z)}^{c-a-b}$ times a polynomial. … ►A small table of zeros is given in Conde and Kalla (1981) and Segura (2008).

5: 19.15 Advantages of Symmetry

… ►Elliptic integrals are special cases of a particular multivariate hypergeometric function called Lauricella’s $F_D$ (Carlson (1961b)). The function $R_{-a}(b_1,b_2,\mathrm{\dots },b_n;z_1,z_2,\mathrm{\dots },z_n)$ (Carlson (1963)) reveals the full permutation symmetry that is partially hidden in $F_D$, and leads to symmetric standard integrals that simplify many aspects of theory, applications, and numerical computation. … ►Symmetry allows the expansion (19.19.7) in a series of elementary symmetric functions that gives high precision with relatively few terms and provides the most efficient method of computing the incomplete integral of the third kind (§19.36(i)). … ►These reduction theorems, unknown in the Legendre theory, allow symbolic integration without imposing conditions on the parameters and the limits of integration (see §19.29(ii)). …

6: 16.21 Differential Equation

… ► $w=G_{p,q}^{m,n}(z;𝐚;𝐛)$ satisfies the differential equation ►

16.21.1 $$\left({(-1)}^{p-m-n}z(\mathit{\vartheta }-a_1+1)\mathrm{\cdots }(\mathit{\vartheta }-a_p+1)-(\mathit{\vartheta }-b_1)\mathrm{\cdots }(\mathit{\vartheta }-b_q)\right)w=0,$$

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Symbols:

$p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters, $w$: solution and $\mathit{\vartheta }$: differential operator

Referenced by:

§16.21, §16.21

Permalink:

http://dlmf.nist.gov/16.21.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.21 and Ch.16

… ►In consequence of (16.19.1) we may assume, without loss of generality, that $p\le q$. With the classification of §16.8(i), when the only singularities of (16.21.1) are a regular singularity at $z=0$ and an irregular singularity at $z=\mathrm{\infty }$. … ►A fundamental set of solutions of (16.21.1) is given by …

7: 5.5 Functional Relations

… ►

5.5.3 $$\mathrm{\Gamma }\left(z\right)\mathrm{\Gamma }\left(1-z\right)=\pi /\sin \left(\pi z\right),$$ $z\ne 0,\pm 1,\mathrm{\dots }$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function and $z$: complex variable

A&S Ref:

6.1.17 (without the condition on $z$.)

Referenced by:

§10.19(i), §15.4(ii), §15.8(ii), (25.9.2), §5.21, (9.10.18), (9.12.15)

Permalink:

http://dlmf.nist.gov/5.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §5.5(ii), §5.5 and Ch.5

►

5.5.4 $$\psi \left(z\right)-\psi \left(1-z\right)=-\pi /\tan \left(\pi z\right),$$ $z\ne 0,\pm 1,\mathrm{\dots }$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $\tan z$: tangent function and $z$: complex variable

A&S Ref:

6.3.7 (without the condition on $z$.)

Permalink:

http://dlmf.nist.gov/5.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §5.5(ii), §5.5 and Ch.5

… ►

5.5.5 $$\mathrm{\Gamma }\left(2z\right)={\pi }^{-1/2}{2}^{2z-1}\mathrm{\Gamma }\left(z\right)\mathrm{\Gamma }\left(z+\frac{1}{2}\right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter and $z$: complex variable

A&S Ref:

6.1.18

Referenced by:

§14.19(ii), §15.4(i), §15.4(iii), (25.9.2), §5.5(iii)

Permalink:

http://dlmf.nist.gov/5.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §5.5(iii), §5.5(iii), §5.5 and Ch.5

… ►

5.5.6 $$\mathrm{\Gamma }\left(nz\right)={(2\pi )}^{(1-n)/2}{n}^{nz-(1/2)}\prod _{k=0}^{n-1}\mathrm{\Gamma }\left(z+\frac{k}{n}\right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $n$: nonnegative integer, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

6.1.20

Referenced by:

§15.4(iii), §5.5(iii)

Permalink:

http://dlmf.nist.gov/5.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §5.5(iii), §5.5(iii), §5.5 and Ch.5

… ►If a positive function $f(x)$ on $(0,\mathrm{\infty })$ satisfies $f(x+1)=xf(x)$, $f(1)=1$, and $\ln f(x)$ is convex (see §1.4(viii)), then $f(x)=\mathrm{\Gamma }\left(x\right)$.

8: 28.15 Expansions for Small $q$

… ►Higher coefficients can be found by equating powers of $q$ in the following continued-fraction equation, with $a={\lambda }_{\nu }\left(q\right)$: ►

28.15.2 $$a-{\nu }^2-\frac{q^2}{a-{(\nu +2)}^2-}\frac{q^2}{a-{(\nu +4)}^2-}\mathrm{\cdots }=\frac{q^2}{a-{(\nu -2)}^2-}\frac{q^2}{a-{(\nu -4)}^2-}\mathrm{\cdots }.$$

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Symbols:

$q=h^2$: parameter, $\nu$: complex parameter and $a$: parameter

Referenced by:

item (e)

Permalink:

http://dlmf.nist.gov/28.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.15(i), §28.15 and Ch.28

… ►

28.15.3 $${\mathrm{me}}_{\nu }(z,q)={\mathrm{e}}^{\mathrm{i}\nu z}-\frac{q}{4}\left(\frac{1}{\nu +1}{\mathrm{e}}^{\mathrm{i}(\nu +2)z}-\frac{1}{\nu -1}{\mathrm{e}}^{\mathrm{i}(\nu -2)z}\right)+\frac{q^2}{32}\left(\frac{1}{(\nu +1)(\nu +2)}{\mathrm{e}}^{\mathrm{i}(\nu +4)z}+\frac{1}{(\nu -1)(\nu -2)}{\mathrm{e}}^{\mathrm{i}(\nu -4)z}-\frac{2({\nu }^2+1)}{{({\nu }^2-1)}^2}{\mathrm{e}}^{\mathrm{i}\nu z}\right)+\mathrm{\cdots };$$

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Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $q=h^2$: parameter, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

20.3.17 (only two terms and without normalization)

Permalink:

http://dlmf.nist.gov/28.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.15(ii), §28.15 and Ch.28

…

9: 8.2 Definitions and Basic Properties

… ►without restrictions on the integration paths. … ►The function ${\gamma }^{\ast }(a,z)$ is entire in $z$ and $a$. … ►(8.2.9) also holds when $a$ is zero or a negative integer, provided that the right-hand side is replaced by its limiting value. …without restriction on $a$. … ►If $w=\gamma (a,z)$ or $\mathrm{\Gamma }(a,z)$, then …

10: Guide to Searching the DLMF

… ►Every page contains a search box in the navigation bar. … ►

a textual word, a number, or a math symbol.

… ►

any LaTeX-like sequence of terms, excluding phrases.

… ►Wildcards allow matching patterns and marking parts of an expression that don’t matter (as for example, which variable name the author uses for a function): … ►If you do not want a term or a sequence of terms in your query to undergo math processing, you should quote them as a phrase. …