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1: 1.9 Calculus of a Complex Variable

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§1.9(vii) Inversion of Limits

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Double Sequences and Series

►A set of complex numbers $\{z_{m,n}\}$ where $m$ and $n$ take all positive integer values is called a double sequence. … ►A double series is the limit of the double sequence … …

2: Guide to Searching the DLMF

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any double-quoted sequence of textual words and numbers.

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3: Bibliography C

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Algorithm 597: Sequence of modified Bessel functions of the first kind. ACM Trans. Math. Software 9 (2), pp. 242–245.

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Notes:

Double-precision Fortran, maximum accuracy 24S.

External Links:

ISSN 0098-3500, Document, GAMS (module 597; package TOMS), ZentralBlatt

Referenced by:

5th item.

Encodings:

BibTeX

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4: 3.8 Nonlinear Equations

… ►Let $z_1,z_2,\mathrm{\dots }$ be a sequence of approximations to a root, or fixed point, $\zeta$. …for all $n$ sufficiently large, where $A$ and $p$ are independent of $n$, then the sequence is said to have convergence of the $p$th order. … ►For real functions $f(x)$ the sequence of approximations to a real zero $\xi$ will always converge (and converge quadratically) if either: … ►As in the case of Table 3.8.1 the quadratic nature of convergence is clearly evident: as the zero is approached, the number of correct decimal places doubles at each iteration. … ►Starting this iteration in the neighborhood of one of the four zeros $\pm 1,\pm \mathrm{i}$, sequences $\{z_n\}$ are generated that converge to these zeros. …

5: 5.17 Barnes’ $G$-Function (Double Gamma Function)

§5.17 Barnes’ $G$-Function (Double Gamma Function)

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5.17.2 $$G\left(n\right)=(n-2)!(n-3)!\mathrm{\cdots }1!,$$ $n=2,3,\mathrm{\dots }$.

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Symbols:

$G\left(z\right)$: Barnes’ $G$-function (or double gamma function), $!$: factorial (as in $n!$) and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/5.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §5.17 and Ch.5

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5.17.3 $$G\left(z+1\right)={(2\pi )}^{z/2}\exp \left(-\frac{1}{2}z(z+1)-\frac{1}{2}\gamma z^2\right)\prod _{k=1}^{\mathrm{\infty }}\left({\left(1+\frac{z}{k}\right)}^k\exp \left(-z+\frac{z^2}{2k}\right)\right).$$

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Symbols:

$G\left(z\right)$: Barnes’ $G$-function (or double gamma function), $\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $k$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/5.17.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §5.17 and Ch.5

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5.17.4 $$\mathrm{Ln}G\left(z+1\right)=\frac{1}{2}z\ln \left(2\pi \right)-\frac{1}{2}z(z+1)+z\mathrm{Ln}\mathrm{\Gamma }\left(z+1\right)-{\int }_0^z\mathrm{Ln}\mathrm{\Gamma }\left(t+1\right)dt.$$

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Symbols:

$G\left(z\right)$: Barnes’ $G$-function (or double gamma function), $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/5.17.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §5.17 and Ch.5

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5.17.6 $$A={\mathrm{e}}^C=\mathrm{1.28242\hspace{0.33em}71291\hspace{0.33em}00622\hspace{0.33em}63687}\mathrm{\dots },$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $A$: Glaisher’s constant and $C$: log of Glaisher’s Constant

Notes:

For more digits see OEIS Sequence A074962; see also Sloane (2003).

Permalink:

http://dlmf.nist.gov/5.17.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §5.17 and Ch.5

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6: Bibliography W

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Algorithm 794: Numerical Hankel transform by the Fortran program HANKEL. ACM Trans. Math. Software 25 (2), pp. 240–250.

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Notes:

Double-precision Fortran, maximum accuracy 10S.

External Links:

ISSN 0098-3500, Document, GAMS (module 794; package TOMS), ZentralBlatt

Referenced by:

7th item, 10th item.

Encodings:

BibTeX

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Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series. Computer Physics Reports 10 (5-6), pp. 189–371.

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External Links:

Document

Referenced by:

§2.11(vi), §3.9(v), §3.9(vi).

Encodings:

BibTeX

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Computation of the Whittaker function of the second kind by summing its divergent asymptotic series with the help of nonlinear sequence transformations. Computers in Physics 10 (5), pp. 496–503.

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External Links:

Document

Referenced by:

§2.11(vi), §2.11(vi).

Encodings:

BibTeX

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Sequence Transformations and their Applications. Mathematics in Science and Engineering, Vol. 154, Academic Press Inc., New York.

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External Links:

ISBN 0-12-757940-0, MathReview (Jan Zítko), ZentralBlatt

Referenced by:

§3.9(vi).

Encodings:

BibTeX

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On the central connection problem for the double confluent Heun equation. Math. Nachr. 195, pp. 267–276.

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External Links:

ISSN 0025-584X, MathReview, ZentralBlatt

Referenced by:

§31.12.

Encodings:

BibTeX

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7: 5.4 Special Values and Extrema

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5.4.2

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Symbols:

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

Referenced by:

§5.4(i), §5.4(i)

Permalink:

http://dlmf.nist.gov/5.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §5.4(i), §5.4 and Ch.5

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5.4.7 $$\mathrm{\Gamma }\left(\frac{1}{3}\right)=\mathrm{2.67893\hspace{0.33em}85347\hspace{0.33em}07747\hspace{0.33em}63365}\mathrm{\dots },$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function

A&S Ref:

6.1.11 (where the value is computed to 10D.)

Notes:

For more digits see OEIS Sequence A073005; see also Sloane (2003).

Permalink:

http://dlmf.nist.gov/5.4.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §5.4(i), §5.4 and Ch.5

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5.4.8 $$\mathrm{\Gamma }\left(\frac{2}{3}\right)=\mathrm{1.35411\hspace{0.33em}79394\hspace{0.33em}26400\hspace{0.33em}41694}\mathrm{\dots },$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function

A&S Ref:

6.1.13 (where the value is computed to 10D.)

Notes:

For more digits see OEIS Sequence A073006; see also Sloane (2003).

Permalink:

http://dlmf.nist.gov/5.4.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §5.4(i), §5.4 and Ch.5

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5.4.9 $$\mathrm{\Gamma }\left(\frac{1}{4}\right)=\mathrm{3.62560\hspace{0.33em}99082\hspace{0.33em}21908\hspace{0.33em}31193}\mathrm{\dots },$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function

A&S Ref:

6.1.10 (where the value is computed to 10D.)

Notes:

For more digits see OEIS Sequence A068466; see also Sloane (2003).

Permalink:

http://dlmf.nist.gov/5.4.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §5.4(i), §5.4 and Ch.5

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5.4.10 $$\mathrm{\Gamma }\left(\frac{3}{4}\right)=\mathrm{1.22541\hspace{0.33em}67024\hspace{0.33em}65177\hspace{0.33em}64512}\mathrm{\dots }.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function

A&S Ref:

6.1.14 (where the value is computed to 10D.)

Notes:

For more digits see OEIS Sequence A068465; see also Sloane (2003).

Permalink:

http://dlmf.nist.gov/5.4.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §5.4(i), §5.4 and Ch.5

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8: 18.33 Polynomials Orthogonal on the Unit Circle

… ►This states that for any sequence ${\{{\alpha }_n\}}_{n=0}^{\mathrm{\infty }}$ with ${\alpha }_n\in ℂ$ and the polynomials ${\mathrm{\Phi }}_n(z)$ generated by the recurrence relations (18.33.23), (18.33.24) with ${\mathrm{\Phi }}_0(z)=1$ satisfy the orthogonality relation (18.33.17) for a unique probability measure $\mu$ with infinite support on the unit circle. … ►

18.33.32

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $\ln z$: principal branch of logarithm function, $w(x)$: weight function and $z$: complex variable

Source:

Simon (2005a, (1.1.19))

Referenced by:

Erratum (V1.2.0) Section 18.33

Permalink:

http://dlmf.nist.gov/18.33.E32

Encodings:

pMML, png, TeX

See also:

Annotations for §18.33(vi), §18.33(vi), §18.33 and Ch.18

9: 4.2 Definitions

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4.2.11 $$\mathrm{e}=\mathrm{2.71828\; 18284\; 59045\; 23536}\mathrm{\dots }$$

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Defines:

$\mathrm{e}$: base of natural logarithm

A&S Ref:

4.2.22 (with 10D value)

Notes:

For more digits see OEIS Sequence A001113; see also Sloane (2003).

Permalink:

http://dlmf.nist.gov/4.2.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §4.2(ii), §4.2 and Ch.4

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4.2.17 $${\log }_{10}\mathrm{e}=\mathrm{0.43429\; 44819\; 03251\; 82765}\mathrm{\dots },$$

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Symbols:

$\mathrm{e}$: base of natural logarithm and ${\log }_{a}z$: logarithm to general base

A&S Ref:

4.1.22 (with 10D value)

Notes:

For more digits see OEIS Sequence A002285; see also Sloane (2003).

Permalink:

http://dlmf.nist.gov/4.2.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §4.2(ii), §4.2 and Ch.4

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4.2.18 $$\ln 10=\mathrm{2.30258\; 50929\; 94045\; 68401}\mathrm{\dots }.$$

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Symbols:

$\ln z$: principal branch of logarithm function

A&S Ref:

4.1.23 (with 10D value.)

Notes:

For more digits see OEIS Sequence A002392; see also Sloane (2003).

Permalink:

http://dlmf.nist.gov/4.2.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §4.2(ii), §4.2 and Ch.4

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4.2.25 $$\exp z=\zeta ⟺z=\mathrm{Ln}\zeta .$$

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Symbols:

$\exp z$: exponential function, $\mathrm{Ln}z$: general logarithm function and $z$: complex variable

A&S Ref:

4.2.4

Permalink:

http://dlmf.nist.gov/4.2.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §4.2(iii), §4.2 and Ch.4

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4.2.35 $$z^a=w⟺z=\exp \left(\frac{1}{a}\mathrm{Ln}w\right).$$

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Defines:

$a\ne 0$: complex constant (locally)

Symbols:

$\exp z$: exponential function, $\mathrm{Ln}z$: general logarithm function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.2.E35

Encodings:

pMML, png, TeX

See also:

Annotations for §4.2(iv), §4.2(iv), §4.2 and Ch.4

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10: Bibliography S

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Coulomb functions analytic in the energy. Comput. Phys. Comm. 25 (1), pp. 87–95.

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Notes:

Double-precision Fortran code.

External Links:

Document, Code

Referenced by:

§33.1, §33.14(iv), 9th item.

Encodings:

BibTeX

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Non-linear transformations of divergent and slowly convergent sequences. J. Math. Phys. 34, pp. 1–42.

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External Links:

MathReview (J. Kuntzmann), ZentralBlatt

Referenced by:

§17.18.

Encodings:

BibTeX

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Algorithm AS 239. Chi-squared and incomplete gamma integral. Appl. Statist. 37 (3), pp. 466–473.

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Notes:

Double-precision Fortran.

External Links:

FullText, Code

Referenced by:

5th item.

Encodings:

BibTeX

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On the double points of a Mathieu equation. J. Comput. Appl. Math. 107 (1), pp. 111–125.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§28.7.

Encodings:

BibTeX

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The On-Line Encyclopedia of Integer Sequences. Notices Amer. Math. Soc. 50 (8), pp. 912–915.

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External Links:

ISSN 0002-9920, FullText, Electronic Database, MathReview (Christian Krattenthaler), ZentralBlatt

Referenced by:

(19.20.2), (19.20.22), (19.30.13), (25.11.40), (3.12.1), (3.12.3), (3.12.4), §3.12, (4.2.11), (4.2.17), (4.2.18), (4.4.19), (5.17.6), (5.2.3), (5.4.10), (5.4.6), (5.4.7), (5.4.8), (5.4.9), (6.13.1).

Encodings:

BibTeX

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