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21: 21.5 Modular Transformations

… ►Let $𝐀$, $𝐁$, $𝐂$, and $𝐃$ be $g\times g$ matrices with integer elements such that ►

21.5.1

ⓘ

Defines:

Matrix $𝚪$ (locally)

Permalink:

http://dlmf.nist.gov/21.5.E1

Encodings:

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See also:

Annotations for §21.5(i), §21.5 and Ch.21

… ►

21.5.4 $$\theta \left({\left[{[𝐂𝛀+𝐃]}^{-1}\right]}^{\mathrm{T}}𝐳\right|[𝐀𝛀+𝐁]{[𝐂𝛀+𝐃]}^{-1})=\xi (𝚪)\sqrt{det[𝐂𝛀+𝐃]}{\mathrm{e}}^{\pi \mathrm{i}𝐳\cdot \left[{[𝐂𝛀+𝐃]}^{-1}𝐂\right]\cdot 𝐳}\theta \left(𝐳\right|𝛀).$$

ⓘ

Symbols:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $det$: determinant, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${𝐀}^{\mathrm{T}}$: transpose of matrix, $𝛀$: a Riemann matrix, Matrix $𝚪$ and $\xi (𝚪)$: eighth root of unity

Referenced by:

§21.5(i)

Permalink:

http://dlmf.nist.gov/21.5.E4

Encodings:

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See also:

Annotations for §21.5(i), §21.5 and Ch.21

… ►

21.5.7

ⓘ

Symbols:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function, $g$: positive integer, $𝛀$: a Riemann matrix, $\mathrm{diag}𝐀$: Transpose of $[A_{11},A_{22},\mathrm{\dots },A_{gg}]$ and Matrix $𝚪$

Referenced by:

(21.5.8), Erratum (V1.0.11) for Clarifications

Permalink:

http://dlmf.nist.gov/21.5.E7

Encodings:

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See also:

Annotations for §21.5(i), §21.5 and Ch.21

… ►

21.5.9 $$\theta \left[\genfrac{}{}{0.0pt}{}{𝐃𝜶-𝐂𝜷+\frac{1}{2}\mathrm{diag}[𝐂{𝐃}^{\mathrm{T}}]}{-𝐁𝜶+𝐀𝜷+\frac{1}{2}\mathrm{diag}[𝐀{𝐁}^{\mathrm{T}}]}\right]\left({\left[{[𝐂𝛀+𝐃]}^{-1}\right]}^{\mathrm{T}}𝐳\right|[𝐀𝛀+𝐁]{[𝐂𝛀+𝐃]}^{-1})=\kappa (𝜶,𝜷,𝚪)\sqrt{det[𝐂𝛀+𝐃]}{\mathrm{e}}^{\pi \mathrm{i}𝐳\cdot \left[{[𝐂𝛀+𝐃]}^{-1}𝐂\right]\cdot 𝐳}\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀),$$

ⓘ

Symbols:

$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics, $\pi$: the ratio of the circumference of a circle to its diameter, $det$: determinant, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${𝐀}^{\mathrm{T}}$: transpose of matrix, $𝛀$: a Riemann matrix, $𝜶$: $g$-dimensional vector, $𝜷$: $g$-dimensional vector, $\mathrm{diag}𝐀$: Transpose of $[A_{11},A_{22},\mathrm{\dots },A_{gg}]$ and Matrix $𝚪$

Permalink:

http://dlmf.nist.gov/21.5.E9

Encodings:

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See also:

Annotations for §21.5(ii), §21.5 and Ch.21

…

22: 15.2 Definitions and Analytical Properties

… ►In general, $F(a,b;c;z)$ does not exist when $c=0,-1,-2,\mathrm{\dots }$. … ►For all values of $c$ … ►

Diverges when $\mathrm{\Re }\left(c-a-b\right)\le -1$.

… ►The principal branch of $𝐅(a,b;c;z)$ is an entire function of $a$, $b$, and $c$. …The same properties hold for $F(a,b;c;z)$, except that as a function of $c$, $F(a,b;c;z)$ in general has poles at $c=0,-1,-2,\mathrm{\dots }$. …

23: Software Index

… ► ►►►►►►►

	Open Source													With Book						Commercial
…
Language	C	Int.	C	Ftn	C	C++	C	Int.	C $\mathrm{\#}$	Int. C	Ftn	Int.	Py	C	C Java	C C++ Ftn	C Ftn Mma	Ftn	Ftn	C Ftn Java	Int.	Int.	Int.	C Ftn
…
7.25(iv) $C\left(x\right)$, $S\left(x\right)$, $\mathrm{f}\left(x\right)$, $\mathrm{g}\left(x\right)$, $x\in ℝ$	✓		✓						✓			a	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓
7.25(v) $C\left(z\right)$, $S\left(z\right)$, $z\in ℂ$	✓											a	✓								✓	✓	✓
…
15.20(ii) ${}_2{}^{}F_1^{}(a,b;c;x)$, $x,a,b,c\in ℝ$	✓		✓		✓	✓						a	✓	✓			✓		✓		✓	✓	a
15.20(iii) ${}_2{}^{}F_1^{}(a,b;c;z)$, $z,a,b,c\in ℂ$	✓											a	✓			✓					✓	✓	a
…

…

24: Bibliography F

… ►

ⓘ

Notes:

The “Freely Distributable LIBM” package provides implementations of standard elementary functions plus a few higher functions, e.g. gamma. Double precision, maximum accuracy 20S. Developed by Sun Microsystems.

External Links:

GAMS (package FDLIBM)

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

BibTeX

… ►

Complex zeros of the error function and of the complementary error function. Math. Comp. 27 (122), pp. 401–407.

ⓘ

External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§7.13(i), §7.13(ii), 1st item.

Encodings:

BibTeX

… ►

Table errata; Complex zeros of Fresnel integrals. Math. Comp. 27 (121), pp. 219.

ⓘ

Notes:

Errata in Abramowitz and Stegun (1964)

External Links:

FullText, MathReview

Referenced by:

(7.13.6), §7.13(iii).

Encodings:

BibTeX

… ►

Spheroidal Wave Functions. Stanford University Press, Stanford, CA.

ⓘ

External Links:

MathReview (J. C. P. Miller), ZentralBlatt

Referenced by:

§30.1, §30.10, 2nd item, Ch.30.

Encodings:

BibTeX

… ►

Numerical treatment of Coulomb wave functions. Rev. Mod. Phys. 27 (4), pp. 399–411.

ⓘ

External Links:

Document, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§33.10(ii), §33.11, §33.12(i), §33.9(ii).

Encodings:

BibTeX

…

25: 26.11 Integer Partitions: Compositions

… ► $c\left(n\right)$ 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 $T=\{2,3,4,\mathrm{\dots }\}$. … ►

26.11.1 $$c\left(0\right)=c(\in T,0)=1.$$

ⓘ

Symbols:

$\in$: element of, $c\left(n\right)$: number of compositions of $n$ , $c(\mathrm{condition},n)$: restricted number of compositions of $n$ and $T$: set

Permalink:

http://dlmf.nist.gov/26.11.E1

Encodings:

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See also:

Annotations for §26.11 and Ch.26

… ►

26.11.2 $$c_m\left(0\right)={\delta }_{0,m},$$

ⓘ

Symbols:

${\delta }_{j,k}$: Kronecker delta, $c_m\left(n\right)$: number of compositions of $n$ into exactly $m$ parts and $m$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.11.E2

Encodings:

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See also:

Annotations for §26.11 and Ch.26

►

26.11.3 $$c_m\left(n\right)=\left(\genfrac{}{}{0.0pt}{}{n-1}{m-1}\right),$$

ⓘ

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $c_m\left(n\right)$: number of compositions of $n$ into exactly $m$ parts, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.11.E3

Encodings:

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See also:

Annotations for §26.11 and Ch.26

…

26: 30.9 Asymptotic Approximations and Expansions

… ►

30.9.4 $${\lambda }_n^m\left({\gamma }^2\right)\sim 2q|\gamma |+c_0+c_1{|\gamma |}^{-1}+c_2{|\gamma |}^{-2}+\mathrm{\cdots },$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, ${\lambda }_n^m\left({\gamma }^2\right)$: eigenvalues of the spheroidal differential equation, $m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter, $q$ and $c_n$: coeffients

A&S Ref:

21.8.2

Referenced by:

item

Permalink:

http://dlmf.nist.gov/30.9.E4

Encodings:

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See also:

Annotations for §30.9(ii), §30.9 and Ch.30

►where ►

$2c_0=-q^2-1+m^2,$

►

$8c_1=-q^3-q+m^{2}q,$

►

$2^6{c}_2=-5q^4-10q^2-1+2m^2(3q^2+1)-m^4,$

…

27: Bibliography T

… ►

Some definite integrals and Fourier series for Jacobian elliptic functions. Z. Angew. Math. Mech. 49, pp. 95–96.

ⓘ

External Links:

ISSN 0044-2267, Document, MathReview (B. C. Carlson), ZentralBlatt

Referenced by:

§22.11.

Encodings:

BibTeX

… ►

The universal Askey-Wilson algebra and DAHA of type $(C_1^{\vee },C_1)$ . SIGMA 9, pp. Paper 047, 40 pp..

ⓘ

External Links:

ISSN 1815-0659, Link, MathReview (Renato Álvarez-Nodarse), ZentralBlatt

Referenced by:

§18.38(iii).

Encodings:

BibTeX

… ►

Eigenfunction Expansions Associated with Second-Order Differential Equations. Clarendon Press, Oxford.

ⓘ

External Links:

MathReview (H. Pollard)

Referenced by:

§1.18(x).

Encodings:

BibTeX

… ►

The Theory of Functions. 2nd edition, Oxford University Press, Oxford.

ⓘ

External Links:

ZentralBlatt

Referenced by:

§1.10(ix), §1.10(v), §1.8(i), §1.8(ii), §1.8(iii), §1.9(vii).

Encodings:

BibTeX

… ►

Une nouvelle représentation explicite des nombres d’Euler. C. R. Acad. Sci. Paris Sér. A-B 286 (19), pp. A807–A809.

ⓘ

External Links:

MathReview, ZentralBlatt

Referenced by:

§24.6.

Encodings:

BibTeX

…

28: Bibliography B

… ►

Evaluation of the incomplete gamma function of imaginary argument by Chebyshev polynomials. Math. Comp. 15 (73), pp. 7–11.

ⓘ

External Links:

FullText, MathReview (J. C. P. Miller), ZentralBlatt

Referenced by:

§8.7.

Encodings:

BibTeX

… ►

Zero-field susceptibility of the two-dimensional Ising model near $T_c$ . Phys. Rev. Lett. 31, pp. 1409–1411.

ⓘ

External Links:

Document

Referenced by:

§32.16.

Encodings:

BibTeX

… ►

Ramanujan’s theories of elliptic functions to alternative bases. Trans. Amer. Math. Soc. 347 (11), pp. 4163–4244.

ⓘ

External Links:

ISSN 0002-9947, FullText, MathReview (J. Borwein), ZentralBlatt

Referenced by:

§15.8(v), §20.11(iii).

Encodings:

BibTeX

… ►

Asymptotics of Stirling numbers of the second kind. Proc. Amer. Math. Soc. 42 (2), pp. 575–580.

ⓘ

Notes:

Erratum: same journal v. 48 (1975), p. 518

External Links:

FullText, MathReview (H. A. Lauwerier), ZentralBlatt

Referenced by:

§26.8(vii).

Encodings:

BibTeX

… ►

Stability of repulsive Bose-Einstein condensates in a periodic potential. Phys. Rev. E (3) 63 (036612), pp. 1–11.

ⓘ

External Links:

Document

Referenced by:

§29.19(i).

Encodings:

BibTeX

…

29: 23.22 Methods of Computation

… ►Suppose that the invariants $g_2=c$, $g_3=d$, are given, for example in the differential equation (23.3.10) or via coefficients of an elliptic curve (§23.20(ii)). … ►

In the general case, given by $cd\ne 0$, we compute the roots $\alpha$, $\beta$, $\gamma$, say, of the cubic equation $4t^3-ct-d=0$; see §1.11(iii). These roots are necessarily distinct and represent $e_1$, $e_2$, $e_3$ in some order.

If $c$ and $d$ are real, and the discriminant is positive, that is $c^3-27d^2>0$, then $e_1$, $e_2$, $e_3$ can be identified via (23.5.1), and $k^2$, $k_{}^{\prime }{}_{}{}^2$ obtained from (23.6.16).

If , or $c$ and $d$ are not both real, then we label $\alpha$, $\beta$, $\gamma$ so that the triangle with vertices $\alpha$, $\beta$, $\gamma$ is positively oriented and $[\alpha ,\gamma ]$ is its longest side (chosen arbitrarily if there is more than one). In particular, if $\alpha$, $\beta$, $\gamma$ are collinear, then we label them so that $\beta$ is on the line segment $(\alpha ,\gamma )$. In consequence, $k^2=(\beta -\gamma )/(\alpha -\gamma )$, $k_{}^{\prime }{}_{}{}^2=(\alpha -\beta )/(\alpha -\gamma )$ satisfy $\mathrm{\Im }{k}^2\ge 0\ge \mathrm{\Im }k_{}^{\prime }{}_{}{}^2$ (with strict inequality unless $\alpha$, $\beta$, $\gamma$ are collinear); also $|k^2|$, $|k_{}^{\prime }{}_{}{}^2|\le 1$.

Finally, on taking the principal square roots of $k^2$ and $k_{}^{\prime }{}_{}{}^2$ we obtain values for $k$ and $k^{\prime }$ that lie in the 1st and 4th quadrants, respectively, and $2{\omega }_1$, $2{\omega }_3$ are given by

23.22.1 $$2{\omega }_{1}M(1,k^{\prime })=-2\mathrm{i}{\omega }_{3}M(1,k)=\frac{\pi }{3}\sqrt{\frac{c(2+k^{2}k_{}^{\prime }{}_{}{}^2)(k_{}^{\prime }{}_{}{}^2-k^2)}{d(1-k^{2}k_{}^{\prime }{}_{}{}^2)}},$$

ⓘ

Symbols:

$M(a,g)$: arithmetic-geometric mean, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators, $k$: modulus and $k^{\prime }$: complementary modulus

Proof sketch:

Combine (23.6.16), (23.6.17), (22.20.6), (23.3.5)–(23.3.7), and use analytical continuation.

Referenced by:

§23.22(ii)

Permalink:

http://dlmf.nist.gov/23.22.E1

Encodings:

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See also:

Annotations for §23.22(ii), §23.22(ii), §23.22 and Ch.23

where $M$ denotes the arithmetic-geometric mean (see §§19.8(i) and 22.20(ii)). This process yields 2 possible pairs ($2{\omega }_1$, $2{\omega }_3$), corresponding to the 2 possible choices of the square root.

►

If $d=0$, then

23.22.2 $$2{\omega }_1=-2\mathrm{i}{\omega }_3=\frac{{\left(\mathrm{\Gamma }\left(\frac{1}{4}\right)\right)}^2}{2\sqrt{\pi }{c}^{1/4}}.$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit and ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators

Referenced by:

§22.20(iv), §23.22(ii)

Permalink:

http://dlmf.nist.gov/23.22.E2

Encodings:

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See also:

Annotations for §23.22(ii), §23.22(ii), §23.22 and Ch.23

There are 4 possible pairs ($2{\omega }_1$, $2{\omega }_3$), corresponding to the 4 rotations of a square lattice. The lemniscatic case occurs when $c>0$ and ${\omega }_1>0$.

►

If $c=0$, then

23.22.3 $$2{\omega }_1=2{\mathrm{e}}^{-\pi \mathrm{i}/3}{\omega }_3=\frac{{\left(\mathrm{\Gamma }\left(\frac{1}{3}\right)\right)}^3}{2\pi d^{1/6}}.$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators

Referenced by:

§23.22(ii)

Permalink:

http://dlmf.nist.gov/23.22.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §23.22(ii), §23.22(ii), §23.22 and Ch.23

There are 6 possible pairs ($2{\omega }_1$, $2{\omega }_3$), corresponding to the 6 rotations of a lattice of equilateral triangles. The equianharmonic case occurs when $d>0$ and ${\omega }_1>0$.

… ►Assume $c=g_2=-4(3-2\mathrm{i})$ and $d=g_3=4(4-2\mathrm{i})$. …

30: 34 3j, 6j, 9j Symbols

…