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1: 23.15 Definitions

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23.15.5 $$f(𝒜\tau )=c_{𝒜}{(c\tau +d)}^{\mathrm{\ell }}f(\tau ),$$ $\mathrm{\Im }\tau >0$,

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Symbols:

$\mathrm{\Im }$: imaginary part, $\tau$: complex variable, $𝒜$: bilinear transformation, $c$: integer, $d$: integer, $f(\tau )$: modular function, $c_{𝒜}$: constant and $\mathrm{\ell }$: level

Permalink:

http://dlmf.nist.gov/23.15.E5

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Annotations for §23.15(i), §23.15 and Ch.23

►where $c_{𝒜}$ is a constant depending only on $𝒜$, and $\mathrm{\ell }$ (the level) is an integer or half an odd integer. (Some references refer to $2\mathrm{\ell }$ as the level). …

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

3: 3.1 Arithmetics and Error Measures

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§3.1(iv) Level-Index Arithmetic

… ►In level-index arithmetic $x$ is represented by $\mathrm{\ell }+a$ (or $-(\mathrm{\ell }+a)$ for negative numbers). … ►For further references on level-index arithmetic (and also other arithmetics) see Anuta et al. (1996). …

4: Mathematical Introduction

… ►In the Handbook this information is grouped at the section level and appears under the heading Sources in the References section. In the DLMF this information is provided in pop-up windows at the subsection level. …

5: Bibliography T

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Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159 (1), pp. 151–174.

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External Links:

ISSN 0010-3616, FullText, MathReview (Estelle L. Basor), ZentralBlatt

Referenced by:

§32.14.

Encodings:

BibTeX

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6: 37.19 Other Orthogonal Polynomials of $d$ Variables

… ►However, when the one-variable OPs are taken from a higher level in the Askey scheme, the analogues of the denominators in the arguments in (37.14.7) will be parameters depending on $x_{\mathrm{\ell }}$ variables. …

7: 18.40 Methods of Computation

… ►Achieving precisions at this level shown above requires higher than normal computational precision, see Gautschi (2009). …

8: Bibliography C

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Level-Index Arithmetic: An Introductory Survey. In Numerical Analysis and Parallel Processing (Lancaster, 1987), P. R. Turner (Ed.), Lecture Notes in Math., Vol. 1397, pp. 95–168.

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External Links:

MathReview Entry, ZentralBlatt

Referenced by:

§3.1(iv).

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9: Bibliography M

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Diatomic molecules according to the wave mechanics. II: Vibrational levels. Phys. Rev., II. Ser. 34, pp. 57–64.

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External Links:

ISSN 0031-899X, ZentralBlatt

Referenced by:

(18.39.16).

Encodings:

BibTeX

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10: Bibliography S

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Decoherent dynamics of a two-level system coupled to a sea of spins. Phys. Rev. Lett. 81 (26), pp. 5710–5713.

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External Links:

Document

Referenced by:

§11.12.

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