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11: 1.10 Functions of a Complex Variable

… ►If we can assign a unique value $f(z)$ to $F(z)$ at each point of $D$, and $f(z)$ is analytic on $D$, then $f(z)$ is a branch of $F(z)$. … ►(b) By specifying the value of $F(z)$ at a point $z_0$ (not a branch point), and requiring $F(z)$ to be continuous on any path that begins at $z_0$ and does not pass through any branch points or other singularities of $F(z)$. … ►Then the value of $F(z)$ at any other point is obtained by analytic continuation. … ►has a unique solution $z=F(w)$ analytic at $w=w_0$, and …in a neighborhood of $w_0$, where $n{F}_n$ is the residue of $1/{(f(z)-f(z_0))}^n$ at $z=z_0$. …

12: 20.4 Values at $z$ = 0

§20.4 Values at $z$ = 0

… ►

20.4.9 $$\frac{{\theta }_2^{\prime \prime }(0,q)}{{\theta }_2(0,q)}=-1-8\sum _{n=1}^{\mathrm{\infty }}\frac{q^{2n}}{{(1+q^{2n})}^2},$$

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Symbols:

${\theta }_j(z,q)$: theta function, $n$: integer and $q$: nome

Permalink:

http://dlmf.nist.gov/20.4.E9

Encodings:

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See also:

Annotations for §20.4(ii), §20.4 and Ch.20

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20.4.10 $$\frac{{\theta }_3^{\prime \prime }(0,q)}{{\theta }_3(0,q)}=-8\sum _{n=1}^{\mathrm{\infty }}\frac{q^{2n-1}}{{(1+q^{2n-1})}^2},$$

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Symbols:

${\theta }_j(z,q)$: theta function, $n$: integer and $q$: nome

Permalink:

http://dlmf.nist.gov/20.4.E10

Encodings:

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See also:

Annotations for §20.4(ii), §20.4 and Ch.20

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20.4.11 $$\frac{{\theta }_4^{\prime \prime }(0,q)}{{\theta }_4(0,q)}=8\sum _{n=1}^{\mathrm{\infty }}\frac{q^{2n-1}}{{(1-q^{2n-1})}^2}.$$

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Symbols:

${\theta }_j(z,q)$: theta function, $n$: integer and $q$: nome

Permalink:

http://dlmf.nist.gov/20.4.E11

Encodings:

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See also:

Annotations for §20.4(ii), §20.4 and Ch.20

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20.4.12 $$\frac{{\theta }_1^{\prime \prime \prime }(0,q)}{{\theta }_1^{\prime }(0,q)}=\frac{{\theta }_2^{\prime \prime }(0,q)}{{\theta }_2(0,q)}+\frac{{\theta }_3^{\prime \prime }(0,q)}{{\theta }_3(0,q)}+\frac{{\theta }_4^{\prime \prime }(0,q)}{{\theta }_4(0,q)}.$$

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Symbols:

${\theta }_j(z,q)$: theta function and $q$: nome

Permalink:

http://dlmf.nist.gov/20.4.E12

Encodings:

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See also:

Annotations for §20.4(ii), §20.4 and Ch.20

13: Bibliography D

… ►

Integralen voor de $\zeta$-functie van Riemann. Mathematica (Zutphen) B5, pp. 170–180 (Dutch).

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Notes:

In Dutch.

External Links:

ZentralBlatt

Referenced by:

(25.5.15), (25.5.16), (25.5.17), (25.5.18), (25.5.19), §25.5(ii).

Encodings:

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Quantum Fields and Strings: A Course for Mathematicians. Vol. 1, 2. American Mathematical Society, Providence, RI.

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Notes:

Material from the Special Year on Quantum Field Theory held at the Institute for Advanced Study, Princeton, NJ, 1996–1997

External Links:

ISBN 0-8218-1198-3, MathReview, ZentralBlatt

Referenced by:

§21.9.

Encodings:

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Zeros of Bernoulli, generalized Bernoulli and Euler polynomials. Mem. Amer. Math. Soc. 73 (386), pp. iv+94.

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External Links:

ISSN 0065-9266, MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§24.12(i), §24.12(iii), §24.16(ii).

Encodings:

BibTeX

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Handbuch der Laplace-Transformation. Bd. II. Anwendungen der Laplace-Transformation. 1. Abteilung. Birkhäuser Verlag, Basel und Stuttgart (German).

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External Links:

MathReview (I. I. Hirschman), ZentralBlatt

Referenced by:

§2.5(i).

Encodings:

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Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.

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Notes:

For a numerical error in one of the zeros see Amos (1985).

External Links:

ISSN 0025-5718, FullText, MathReview, ZentralBlatt

Referenced by:

2nd item.

Encodings:

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14: 16.9 Zeros

… ►Then ${}_p{}^{}F_p^{}(𝐚;𝐛;z)$ has at most finitely many zeros if and only if the $a_j$ can be re-indexed for $j=1,\mathrm{\dots },p$ in such a way that $a_j-b_j$ is a nonnegative integer. … ►Then ${}_p{}^{}F_p^{}(𝐚;𝐛;z)$ has at most finitely many real zeros. …

15: Bibliography

… ►

Some determinants of Bernoulli, Euler and related numbers. Portugal. Math. 18, pp. 91–99.

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External Links:

FullText, MathReview (D. P. Banerjee), ZentralBlatt

Referenced by:

§24.14(ii).

Encodings:

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Sturm-Liouville Theory. Birkhäuser Verlag, Basel.

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Notes:

Past and present, Including papers from the International Colloquium held at the University of Geneva, Geneva, September 15–19, 2003

External Links:

ISBN 978-3-7643-7066-4; 3-7643-7066-1, Document, Link, MathReview (Anton Zettl)

Referenced by:

§1.13(viii), §1.18(x).

Encodings:

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Classical Orthogonal Polynomials. In Orthogonal Polynomials and Applications, C. Brezinski, A. Draux, A. P. Magnus, P. Maroni, and A. Ronveaux (Eds.), Lecture Notes in Math., Vol. 1171, pp. 36–62.

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External Links:

MathReview, ZentralBlatt

Referenced by:

(18.27.12_5), (18.27.6).

Encodings:

BibTeX

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Note on the trivial zeros of Dirichlet $L$-functions. Proc. Amer. Math. Soc. 94 (1), pp. 29–30.

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External Links:

ISSN 0002-9939, FullText, MathReview (Jerzy Kaczorowski), ZentralBlatt

Referenced by:

(25.15.7), (25.15.8), §25.15(ii), §25.15.

Encodings:

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Bernoulli’s power-sum formulas revisited. Math. Gaz. 90 (518), pp. 276–279.

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Referenced by:

§24.4(iii).

Encodings:

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16: 7.3 Graphics

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Figure 7.3.2: Dawson’s integral $F\left(x\right)$, $-3.5\le x\le 3.5$. Magnify

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Figure 7.3.3: Fresnel integrals $C\left(x\right)$ and $S\left(x\right)$, $0\le x\le 4$. Magnify

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Figure 7.3.4: ${|ℱ\left(x\right)|}^2$, $-8\le x\le 8$. Fresnel (1818) introduced the integral $ℱ\left(x\right)$ in his study of the interference pattern at the edge of a shadow. … Magnify

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17: 26.6 Other Lattice Path Numbers

… ►

Delannoy Number $D(m,n)$

► $D(m,n)$ is the number of paths from $(0,0)$ to $(m,n)$ that are composed of directed line segments of the form $(1,0)$, $(0,1)$, or $(1,1)$. … ► $N(n,k)$ is the number of lattice paths from $(0,0)$ to $(n,n)$ that stay on or above the line $y=x$, are composed of directed line segments of the form $(1,0)$ or $(0,1)$, and for which there are exactly $k$ occurrences at which a segment of the form $(0,1)$ is followed by a segment of the form $(1,0)$. … ►

26.6.12 $$C\left(n\right)=\sum _{k=1}^{n}N(n,k),$$

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Symbols:

$C\left(n\right)$: Catalan number, $k$: nonnegative integer, $n$: nonnegative integer and $N(n,k)$: Narayana number

Permalink:

http://dlmf.nist.gov/26.6.E12

Encodings:

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See also:

Annotations for §26.6(iv), §26.6 and Ch.26

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26.6.13 $$M(n)=\sum _{k=0}^n{(-1)}^k\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)C\left(n+1-k\right),$$

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Symbols:

$C\left(n\right)$: Catalan number, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: nonnegative integer, $n$: nonnegative integer and $M(n)$: Motzkin number

Permalink:

http://dlmf.nist.gov/26.6.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §26.6(iv), §26.6 and Ch.26

…

18: Bibliography H

… ►

Certain sums that contain cylindrical functions. Bul. Akad. Štiince RSS Moldoven. 1972 (3), pp. 75–77, 94 (Russian).

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External Links:

MathReview (D. P. Gupta)

Referenced by:

§7.14(iii).

Encodings:

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The Laplace transform for expressions that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1973 (2), pp. 78–80, 93 (Russian).

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External Links:

MathReview (L. Berg)

Referenced by:

§7.14(iii).

Encodings:

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Expansions for the probability function in series of Čebyšev polynomials and Bessel functions. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 77–80, 96 (Russian).

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External Links:

MathReview (V. L. Makarov)

Referenced by:

§7.14(iii).

Encodings:

BibTeX

… ►

Coulomb Wave Functions. In Handbuch der Physik, Bd. 41/1, S. Flügge (Ed.), pp. 408–465.

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External Links:

MathReview (G. Källén)

Referenced by:

§33.2(iii), §33.23(vii), §33.23(vii), §33.24, §33.5(ii), §33.7, Ch.33.

Encodings:

BibTeX

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Variable precision exponential function. ACM Trans. Math. Software 12 (2), pp. 79–91.

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External Links:

ISSN 0098-3500, Document, MathReview Entry, ZentralBlatt

Referenced by:

§4.48(iii).

Encodings:

BibTeX

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19: 15.3 Graphics

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Figure 15.3.1: $F(\frac{4}{3},\frac{9}{16};\frac{14}{5};x),-100\le x\le 1$. Magnify

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Figure 15.3.2: $F(5,-10;1;x),-0.023\le x\le 1$. Magnify

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Figure 15.3.3: $F(1,-10;10;x),-3\le x\le 1$. Magnify

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Figure 15.3.5: $F(\frac{4}{3},\frac{9}{16};\frac{14}{5};x+\mathrm{i}y),0\le x\le 2,-0.5\le y\le 0.5$. … Magnify 3D Help

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Figure 15.3.6: $F(-3,\frac{3}{5};u+\mathrm{i}v;\frac{1}{2}),-6\le u\le 2,-2\le v\le 2$. (With $c=u+\mathrm{i}v$ the only poles occur at $c=0,-1,-2$; compare §15.2(ii).) Magnify 3D Help

…

20: 19.2 Definitions

… ►Let $s^2(t)$ be a cubic or quartic polynomial in $t$ with simple zeros, and let $r(s,t)$ be a rational function of $s$ and $t$ containing at least one odd power of $s$. … ►The integral for $E(\varphi ,k)$ is well defined if $k^2={\sin }^2\varphi =1$, and the Cauchy principal value (§1.4(v)) of $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ is taken if $1-{\alpha }^2{\sin }^2\varphi$ vanishes at an interior point of the integration path. … ►with a branch point at $k=0$ and principal branch $|\mathrm{ph}k|\le \pi$. … ►

§19.2(iv) A Related Function: $R_C(x,y)$

… ►For the special cases of $R_C(x,x)$ and $R_C(0,y)$ see (19.6.15). …