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1: 23.15 Definitions

… ►Also $𝒜$ denotes a bilinear transformation on $\tau$, given by …The set of all bilinear transformations of this form is denoted by SL $(2,\mathbb{Z})$ (Serre (1973, p. 77)). ►A modular function $f(\tau )$ is a function of $\tau$ that is meromorphic in the half-plane $\mathrm{\Im }\tau >0$, and has the property that for all $𝒜\in \text{<span class="ltx\_text hilite">SL</span>}(2,\mathbb{Z})$, or for all $𝒜$ belonging to a subgroup of SL $(2,\mathbb{Z})$, …(Some references refer to $2\mathrm{\ell }$ as the level). …

2: 1.14 Integral Transforms

§1.14 Integral Transforms

… ►When $f$ is real and $\sigma =\frac{1}{2}$, … ►where $A_p=\tan \left(\frac{1}{2}\pi /p\right)$ when , or $\cot \left(\frac{1}{2}\pi /p\right)$ when $p\ge 2$. These bounds are sharp, and equality holds when $p=2$. ►

Fourier Transform

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3: 23.18 Modular Transformations

§23.18 Modular Transformations

… ►and $\lambda \left(\tau \right)$ is a cusp form of level zero for the corresponding subgroup of SL $(2,\mathbb{Z})$. … ► $J\left(\tau \right)$ is a modular form of level zero for SL $(2,\mathbb{Z})$. … ►

23.18.5 $$\eta \left(𝒜\tau \right)=\epsilon (𝒜){\left(-\mathrm{i}(c\tau +d)\right)}^{1/2}\eta \left(\tau \right),$$

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Symbols:

$\eta \left(\tau \right)$: Dedekind’s eta function (or Dedekind modular function), $\mathrm{i}$: imaginary unit, $\tau$: complex variable, $𝒜$: bilinear transformation, $c$: integer, $d$: integer and $\epsilon (𝒜)$: function

Permalink:

http://dlmf.nist.gov/23.18.E5

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See also:

Annotations for §23.18, §23.18 and Ch.23

… ►Note that $\eta \left(\tau \right)$ is of level $\frac{1}{2}$. …

4: 15.17 Mathematical Applications

… ►The logarithmic derivatives of some hypergeometric functions for which quadratic transformations exist (§15.8(iii)) are solutions of Painlevé equations. … ►First, as spherical functions on noncompact Riemannian symmetric spaces of rank one, but also as associated spherical functions, intertwining functions, matrix elements of SL $(2,ℝ)$, and spherical functions on certain nonsymmetric Gelfand pairs. Harmonic analysis can be developed for the Jacobi transform either as a generalization of the Fourier-cosine transform (§1.14(ii)) or as a specialization of a group Fourier transform. … ►Quadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean (§19.22(ii)). … ►By considering, as a group, all analytic transformations of a basis of solutions under analytic continuation around all paths on the Riemann sheet, we obtain the monodromy group. …

5: Bibliography F

… ►

Un théorème de Paley-Wiener pour la transformation de Fourier sur un espace riemannien symétrique de rang un. J. Funct. Anal. 49 (2), pp. 230–268.

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External Links:

ISSN 0022-1236, Document, Link, MathReview (Mogens Flensted-Jensen), ZentralBlatt

Referenced by:

§18.39(i).

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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

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Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).

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External Links:

MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§6.15.

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The transformation properties of the sixth Painlevé equation and one-parameter families of solutions. Lett. Nuovo Cimento (2) 30 (17), pp. 539–544.

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External Links:

ISSN 0024-1318, MathReview (D. A. Lutz)

Referenced by:

§32.10(vi).

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On the asymptotic expansion of Mellin transforms. SIAM J. Math. Anal. 18 (1), pp. 273–282.

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External Links:

ISSN 0036-1410, Document, Link, MathReview (Archibald Brown), ZentralBlatt

Referenced by:

§2.3(vi).

Encodings:

BibTeX

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6: 26.12 Plane Partitions

… ►The number of self-complementary plane partitions in $B(2r,2s,2t)$ is …in $B(2r+1,2s,2t)$ it is …in $B(2r+1,2s+1,2t)$ it is … ►The number of symmetric self-complementary plane partitions in $B(2r,2r,2t)$ is …in $B(2r+1,2r+1,2t)$ it is …

7: Bibliography G

… ►

A new application of the discrete Laguerre polynomials in the numerical evaluation of the Hankel transform of a strongly decreasing even function. J. Comput. Phys. 42 (2), pp. 277–287.

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External Links:

Document, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

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A continued fraction algorithm for the computation of higher transcendental functions in the complex plane. Math. Comp. 21 (97), pp. 18–29.

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External Links:

FullText, MathReview (D. C. Gilles), ZentralBlatt

Referenced by:

§10.74(v), §3.10(ii).

Encodings:

BibTeX

… ►

Algorithm 236: Bessel functions of the first kind. Comm. ACM 7 (8), pp. 479–480.

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Notes:

For certification see same journal 8(1965), pp. 105–106, for remark see ACM Trans. Math. Software, 1(1975), pp. 282–284. Includes Algol program, maximum accuracy: 11D.

External Links:

ISSN 0001-0782, Document

Referenced by:

§10.77(iii).

Encodings:

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Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

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Notes:

Includes Fortran program claiming 12S accuracy

External Links:

ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt

Referenced by:

3rd item, §10.77(ix).

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Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.

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External Links:

ISSN 0003-4916, Link, MathReview (A. O. Barut), ZentralBlatt

Referenced by:

§18.38(iii).

Encodings:

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8: Bibliography L

… ►

Algorithm 917: complex double-precision evaluation of the Wright $\omega$ function. ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.

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External Links:

ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§4.13, 2nd item, §4.48(iv).

Encodings:

BibTeX

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An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.

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External Links:

MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§24.11.

Encodings:

BibTeX

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Eine Verallgemeinerung der Sphäroidfunktionen. Arch. Math. 11, pp. 29–39.

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External Links:

ISSN 0003-9268, Document, MathReview (M. J. O. Strutt), ZentralBlatt

Referenced by:

§30.12.

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Lie algebraic approaches to classical partition identities. Adv. in Math. 29 (1), pp. 15–59.

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External Links:

ISSN 0001-8708, Document, MathReview (S. I. Gel’fand), ZentralBlatt

Referenced by:

§17.16.

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Evaluating infinite integrals involving products of Bessel functions of arbitrary order. J. Comput. Appl. Math. 64 (3), pp. 269–282.

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External Links:

ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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9: Bibliography B

… ►

Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.

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External Links:

Document

Referenced by:

§33.22(iv).

Encodings:

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Transformations of generalized hypergeometric series. Proc. London Math. Soc. (2) 29 (2), pp. 495–502.

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External Links:

Document, ZentralBlatt

Referenced by:

§16.6.

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Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.

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Notes:

Double-precision Fortran code.

External Links:

Document, Code

Referenced by:

1st item, 4th item.

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Quasinormal ringing of Kerr black holes: The excitation factors. Phys. Rev. D 74 (104020), pp. 1–27.

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External Links:

ISSN 0556-2821, Document, MathReview

Referenced by:

6th item.

Encodings:

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Über Sturm-Liouvillesche Polynomsysteme. Math. Z. 29 (1), pp. 730–736.

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External Links:

ISSN 0025-5874, Link, MathReview Entry, ZentralBlatt

Referenced by:

item 1..

Encodings:

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10: 24.2 Definitions and Generating Functions

… ►

$B_{2n+1}=0$ ,

►

${(-1)}^{n+1}{B}_{2n}>0$ , $n=1,2,\mathrm{\dots }$.

… ►

$E_{2n+1}=0$ ,

►

${(-1)}^n{E}_{2n}>0$ .

… ►

24.2.9 $$E_n=2^n{E}_n\left(\frac{1}{2}\right)=\text{integer},$$

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Symbols:

$E_n$: Euler numbers, $E_n\left(x\right)$: Euler polynomials and $n$: integer

A&S Ref:

23.1.2

Permalink:

http://dlmf.nist.gov/24.2.E9

Encodings:

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See also:

Annotations for §24.2(ii), §24.2 and Ch.24

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