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21: Foreword

… ►In 1964 the National Institute of Standards and Technology published the Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, edited by Milton Abramowitz and Irene A. … The success of the original handbook, widely referred to as “Abramowitz and Stegun” (“A&S”), derived not only from the fact that it provided critically useful scientific data in a highly accessible format, but also because it served to standardize definitions and notations for special functions. The provision of standard reference data of this type is a core function of NIST. … ►However, we have also seen the birth of a new age of computing technology, which has not only changed how we utilize special functions, but also how we communicate technical information. …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. …

22: Bille C. Carlson

… ►In his paper Lauricella’s hypergeometric function $F_D$ (1963), he defined the $R$-function, a multivariate hypergeometric function that is homogeneous in its variables, each variable being paired with a parameter. …Also, the homogeneity of the $R$-function has led to a new type of mean value for several variables, accompanied by various inequalities. … ►In Symmetry in c, d, n of Jacobian elliptic functions (2004) he found a previously hidden symmetry in relations between Jacobian elliptic functions, which can now take a form that remains valid when the letters c, d, and n are permuted. … ►Carlson served as a Validator for the original release and publication in May 2010 of the NIST Digital Library of Mathematical Functions and the NIST Handbook of Mathematical Functions.

23: 16 Generalized Hypergeometric Functions & Meijer G-Function

Chapter 16 Generalized Hypergeometric Functions and Meijer $G$-Function

24: 4.25 Continued Fractions

… ►

4.25.1 $$\tan z=\frac{z}{1-}\frac{z^2}{3-}\frac{z^2}{5-}\frac{z^2}{7-}\mathrm{\cdots },$$ $z\ne \pm \frac{1}{2}\pi$, $\pm \frac{3}{2}\pi$, $\mathrm{\dots }$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\tan z$: tangent function and $z$: complex variable

A&S Ref:

4.3.94

Permalink:

http://dlmf.nist.gov/4.25.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.25 and Ch.4

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4.25.2 $$\tan \left(az\right)=\frac{a\tan z}{1+}\frac{(1-a^2){\tan }^{2}z}{3+}\frac{(4-a^2){\tan }^{2}z}{5+}\frac{(9-a^2){\tan }^{2}z}{7+}\mathrm{\cdots },$$ , $az\ne \pm \frac{1}{2}\pi ,\pm \frac{3}{2}\pi ,\mathrm{\dots }$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Re }$: real part, $\tan z$: tangent function, $a$: real or complex constant and $z$: complex variable

A&S Ref:

4.3.95

Permalink:

http://dlmf.nist.gov/4.25.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.25 and Ch.4

… ►

4.25.5 $${\mathrm{e}}^{2a\arctan \left(1/z\right)}=1+\frac{2a}{z-a+}\frac{a^2+1}{3z+}\frac{a^2+4}{5z+}\frac{a^2+9}{7z+}\mathrm{\cdots },$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $\arctan z$: arctangent function, $a$: real or complex constant and $z$: complex variable

A&S Ref:

4.2.42

Permalink:

http://dlmf.nist.gov/4.25.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.25 and Ch.4

…

25: 7.16 Generalized Error Functions

… ►These functions can be expressed in terms of the incomplete gamma function $\gamma (a,z)$ (§8.2(i)) by change of integration variable.

26: 19.3 Graphics

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Figure 19.3.3: $F(\varphi ,k)$ as a function of $k^2$ and ${\sin }^2\varphi$ for $-1\le k^2\le 2$, $0\le {\sin }^2\varphi \le 1$. … Magnify 3D Help

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Figure 19.3.4: $E(\varphi ,k)$ as a function of $k^2$ and ${\sin }^2\varphi$ for $-1\le k^2\le 2$, $0\le {\sin }^2\varphi \le 1$. … Magnify 3D Help

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Figure 19.3.5: $\mathrm{\Pi }({\alpha }^2,k)$ as a function of $k^2$ and ${\alpha }^2$ for , $-2\le {\alpha }^2\le 2$. … Magnify 3D Help

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Figure 19.3.7: $K\left(k\right)$ as a function of complex $k^2$ for $-2\le \mathrm{\Re }\left(k^2\right)\le 2$, $-2\le \mathrm{\Im }\left(k^2\right)\le 2$. … Magnify 3D Help

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Figure 19.3.8: $E\left(k\right)$ as a function of complex $k^2$ for $-2\le \mathrm{\Re }\left(k^2\right)\le 2$, $-2\le \mathrm{\Im }\left(k^2\right)\le 2$. … Magnify 3D Help

…

27: 35.3 Multivariate Gamma and Beta Functions

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35.3.1 $${\mathrm{\Gamma }}_m\left(a\right)={\int }_{𝛀}\mathrm{etr}\left(-𝐗\right){\left|𝐗\right|}^{a-\frac{1}{2}(m+1)}d𝐗,$$ $\mathrm{\Re }\left(a\right)>\frac{1}{2}(m-1)$.

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Symbols:

$\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $𝛀$: space of positive-definite real symmetric $m\times m$ matrices and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.3.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §35.3(i), §35.3 and Ch.35

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35.3.3 $\mathrm{\Re }\left(a\right),\mathrm{\Re }\left(b\right)>\frac{1}{2}(m-1)$.

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Defines:

${\mathrm{B}}_m(a,b)$: multivariate beta function

Symbols:

$\int$: integral, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $b$: complex variable, : less-than for matrices, $m$: positive integer and $𝟎$: $m\times m$ matrix of zeros

Keywords:

definition, multivariate beta function

Permalink:

http://dlmf.nist.gov/35.3.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §35.3(i), §35.3 and Ch.35

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35.3.4 $${\mathrm{\Gamma }}_m\left(a\right)={\pi }^{m(m-1)/4}\prod _{j=1}^m\mathrm{\Gamma }\left(a-\frac{1}{2}(j-1)\right).$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/35.3.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §35.3(ii), §35.3 and Ch.35

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35.3.6 $${\mathrm{\Gamma }}_m\left(a,\mathrm{\dots },a\right)={\mathrm{\Gamma }}_m\left(a\right).$$

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Symbols:

${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $a$: complex variable and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.3.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §35.3(ii), §35.3 and Ch.35

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35.3.7 $${\mathrm{B}}_m(a,b)=\frac{{\mathrm{\Gamma }}_m\left(a\right){\mathrm{\Gamma }}_m\left(b\right)}{{\mathrm{\Gamma }}_m\left(a+b\right)}.$$

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Symbols:

${\mathrm{B}}_m(a,b)$: multivariate beta function, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $a$: complex variable, $b$: complex variable and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.3.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §35.3(ii), §35.3 and Ch.35

…

28: Bibliography G

… ►

Computational Methods in Special Functions – A Survey. In Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), R. A. Askey (Ed.), pp. 1–98. Math. Res. Center, Univ. Wisconsin Publ., No. 35.

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External Links:

MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§19.36(ii), §19.5.

Encodings:

BibTeX

… ►

Problem 72-21, Laplace transforms of Airy functions. SIAM Rev. 15 (4), pp. 796–798.

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Notes:

Problem proposed by P. Smith.

External Links:

Document

Referenced by:

(9.10.14), (9.10.15), (9.10.16), §9.10(v).

Encodings:

BibTeX

… ►

The Applications of Elliptic Functions. MacMillan, London.

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Notes:

Reprinted by Dover Publications Inc., New York, 1959.

External Links:

Link, ZentralBlatt

Referenced by:

§19.30(iii), Ch.19.

Encodings:

BibTeX

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The Applications of Elliptic Functions. Dover Publications Inc., New York.

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Notes:

Reprint of Macmillan, London edition, 1892

External Links:

MathReview, ZentralBlatt

Referenced by:

§22.18(iii), §22.19(iv), §22.8(iii).

Encodings:

BibTeX

… ►

Mathieu functions, a visual approach. Amer. J. Phys. 71 (3), pp. 233–242.

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External Links:

Document

Referenced by:

§28.33(ii).

Encodings:

BibTeX

…

29: Bibliography Y

… ►

The calculation of the Riemann zeta function in the complex domain. USSR Comput. Math. and Math. Phys. 25 (2), pp. 111–119.

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External Links:

Document, ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

BibTeX

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Computation of the derivatives of the Riemann zeta-function in the complex domain. USSR Comput. Math. and Math. Phys. 28 (4), pp. 115–124.

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Notes:

Includes Fortran programs.

External Links:

Document, ZentralBlatt

Referenced by:

§25.18(i), §25.21(iii).

Encodings:

BibTeX

… ►

Computation of Kummer functions $U(a,b,x)$ for large argument $x$ by using the $\tau$-method. Trans. Inform. Process. Soc. Japan 36 (10), pp. 2335–2342 (Japanese).

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Notes:

Japanese with English summary. Double-precision Fortran.

External Links:

ISSN 0387-5806, FullText, MathReview

Referenced by:

§13.32(ii).

Encodings:

BibTeX

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Coulomb wave functions in repulsive fields. Phys. Rev. 49 (2), pp. 174–189.

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External Links:

Document, ZentralBlatt

Referenced by:

§33.10(i), §33.2(ii), §33.2(iii), §33.2(iv), §33.5(i), §33.5(iv), §33.9(ii).

Encodings:

BibTeX

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Bessel Functions. Part IV: Kelvin Functions. Royal Society Mathematical Tables, Volume 10, Cambridge University Press, Cambridge-New York.

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External Links:

MathReview (John Todd), ZentralBlatt

Referenced by:

§10.65(ii), §10.68(iii), §10.68(iv), §10.71(i), 1st item.

Encodings:

BibTeX

…

30: 7.19 Voigt Functions

… ►

7.19.1 $$𝖴(x,t)=\frac{1}{\sqrt{4\pi t}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-{(x-y)}^2/(4t)}}{1+y^2}dy,$$

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Defines:

$𝖴(x,t)$: Voigt function

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $x$: real variable

Referenced by:

§7.19(iii)

Permalink:

http://dlmf.nist.gov/7.19.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §7.19(i), §7.19 and Ch.7

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7.19.2 $$𝖵(x,t)=\frac{1}{\sqrt{4\pi t}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{y{\mathrm{e}}^{-{(x-y)}^2/(4t)}}{1+y^2}dy.$$

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Defines:

$𝖵(x,t)$: Voigt function

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $x$: real variable

Referenced by:

§7.19(iii)

Permalink:

http://dlmf.nist.gov/7.19.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §7.19(i), §7.19 and Ch.7

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7.19.3 $$𝖴(x,t)+\mathrm{i}𝖵(x,t)=\sqrt{\frac{\pi }{4t}}{\mathrm{e}}^{z^2}\mathrm{erfc}z,$$ $z=(1-\mathrm{i}x)/(2\sqrt{t})$.

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Symbols:

$𝖴(x,t)$: Voigt function, $𝖵(x,t)$: Voigt function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable and $z$: complex variable

Referenced by:

§7.19(i), §7.22(iv)

Permalink:

http://dlmf.nist.gov/7.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §7.19(i), §7.19 and Ch.7

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7.19.4 $$H(a,u)=\frac{a}{\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-t^2}dt}{{(u-t)}^2+a^2}=\frac{1}{a\sqrt{\pi }}𝖴(\frac{u}{a},\frac{1}{4a^2}).$$

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Defines:

$H(a,u)$: line-broadening function

Symbols:

$𝖴(x,t)$: Voigt function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm and $\int$: integral

Permalink:

http://dlmf.nist.gov/7.19.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §7.19(i), §7.19 and Ch.7

► $H(a,u)$ is sometimes called the line broadening function; see, for example, Finn and Mugglestone (1965). …