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11: 23.15 Definitions

… ►Also $𝒜$ denotes a bilinear transformation on $\tau$, given by ►

23.15.3 $$𝒜\tau =\frac{a\tau +b}{c\tau +d},$$

ⓘ

Symbols:

$\tau$: complex variable, $𝒜$: bilinear transformation, $a$: integer, $b$: integer, $c$: integer and $d$: integer

Referenced by:

§20.11(i), §23.18

Permalink:

http://dlmf.nist.gov/23.15.E3

Encodings:

pMML, png, TeX

See also:

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

… ►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 $𝒜\in \text{SL}(2,\mathbb{Z})$, or for all $𝒜$ belonging to a subgroup of SL$(2,\mathbb{Z})$, ►

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

ⓘ

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

Encodings:

pMML, png, TeX

See also:

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. …

12: 32.7 Bäcklund Transformations

… ►Then the transformations …and … ►Then … ►Then … ►Then …

13: 21.6 Products

… ►

21.6.1 $$𝒦={\mathbb{Z}}^{g\times h}𝐓/({\mathbb{Z}}^{g\times h}𝐓\cap {\mathbb{Z}}^{g\times h}),$$

ⓘ

Defines:

$𝒦$: set of matrices (locally)

Symbols:

$\mathbb{Z}$: set of all integers, $\cap$: intersection, $g$: positive integer and $h$: positive integer

Permalink:

http://dlmf.nist.gov/21.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §21.6(i), §21.6 and Ch.21

►that is, $𝒦$ is the set of all $g\times h$ matrices that are obtained by premultiplying $𝐓$ by any $g\times h$ matrix with integer elements; two such matrices in $𝒦$ are considered equivalent if their difference is a matrix with integer elements. … ►

21.6.2 $$𝒟=|{𝐓}^{\mathrm{T}}{\mathbb{Z}}^h/({𝐓}^{\mathrm{T}}{\mathbb{Z}}^h\cap {\mathbb{Z}}^h)|,$$

ⓘ

Defines:

$𝒟$: number of elements (locally)

Symbols:

$\mathbb{Z}$: set of all integers, $\cap$: intersection, ${𝐀}^{\mathrm{T}}$: transpose of matrix and $h$: positive integer

Permalink:

http://dlmf.nist.gov/21.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §21.6(i), §21.6 and Ch.21

►that is, $𝒟$ is the number of elements in the set containing all $h$-dimensional vectors obtained by multiplying ${𝐓}^{\mathrm{T}}$ on the right by a vector with integer elements. … ►

21.6.3 $$\prod _{j=1}^h\theta \left(\sum _{k=1}^h{T}_{jk}{𝐳}_k\right|𝛀)=\frac{1}{{𝒟}^g}\sum _{𝐀\in 𝒦}\sum _{𝐁\in 𝒦}{\mathrm{e}}^{2\pi \mathrm{i}\mathrm{tr}\left[\frac{1}{2}{𝐀}^{\mathrm{T}}𝛀𝐀+{𝐀}^{\mathrm{T}}[𝐙+𝐁]\right]}\prod _{j=1}^h\theta \left({𝐳}_j+𝛀{𝐚}_j+{𝐛}_j\right|𝛀),$$

ⓘ

Symbols:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{tr}𝐀$: trace of matrix, ${𝐀}^{\mathrm{T}}$: transpose of matrix, $g$: positive integer, $h$: positive integer, $𝛀$: a Riemann matrix, $T_{jk}$: element, $𝒦$: set of matrices and $𝒟$: number of elements

Permalink:

http://dlmf.nist.gov/21.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §21.6(i), §21.6 and Ch.21

…

14: 10.44 Sums

… ►

10.44.1 $${𝒵}_{\nu }\left(\lambda z\right)={\lambda }^{\pm \nu }\sum _{k=0}^{\mathrm{\infty }}\frac{{({\lambda }^2-1)}^k{(\frac{1}{2}z)}^k}{k!}{𝒵}_{\nu \pm k}\left(z\right),$$ .

ⓘ

Symbols:

$!$: factorial (as in $n!$), ${𝒵}_{\nu }\left(z\right)$: modified cylinder function, $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.51

Referenced by:

§10.44(i), §10.66

Permalink:

http://dlmf.nist.gov/10.44.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.44(i), §10.44 and Ch.10

►If $𝒵=I$ and the upper signs are taken, then the restriction on $\lambda$ is unnecessary. … ►

10.44.3 $${𝒵}_{\nu }\left(u\pm v\right)=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{(\pm 1)}^k{𝒵}_{\nu +k}\left(u\right)I_k\left(v\right),$$ .

ⓘ

Symbols:

$I_{\nu }\left(z\right)$: modified Bessel function of the first kind, ${𝒵}_{\nu }\left(z\right)$: modified cylinder function, $k$: nonnegative integer and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.44.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.44(ii), §10.44(ii), §10.44 and Ch.10

►The restriction is unnecessary when $𝒵=I$ and $\nu$ is an integer. …

15: 10.50 Wronskians and Cross-Products

… ►

$$𝒲\left\{{𝗃}_n\left(z\right),{𝗒}_n\left(z\right)\right\}=z^{-2},$$

►

$$𝒲\left\{{𝗁}_n^{(1)}\left(z\right),{𝗁}_n^{(2)}\left(z\right)\right\}=-2\mathrm{i}{z}^{-2}.$$

►

$$𝒲\left\{{𝗂}_n^{(1)}\left(z\right),{𝗂}_n^{(2)}\left(z\right)\right\}={(-1)}^{n+1}{z}^{-2},$$

►

$$𝒲\left\{{𝗂}_n^{(1)}\left(z\right),{𝗄}_n\left(z\right)\right\}=𝒲\left\{{𝗂}_n^{(2)}\left(z\right),{𝗄}_n\left(z\right)\right\}=-\frac{1}{2}\pi z^{-2}.$$

…

16: 1.16 Distributions

… ►We denote it by $𝒟(I)$. … ► … ►for all $\varphi \in 𝒯$. … ► …

17: 26.21 Tables

… ►Abramowitz and Stegun (1964, Chapter 24) tabulates binomial coefficients $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$ for $m$ up to 50 and $n$ up to 25; extends Table 26.4.1 to $n=10$; tabulates Stirling numbers of the first and second kinds, $s(n,k)$ and $S(n,k)$, for $n$ up to 25 and $k$ up to $n$; tabulates partitions $p\left(n\right)$ and partitions into distinct parts $p(𝒟,n)$ for $n$ up to 500. …

18: 10.66 Expansions in Series of Bessel Functions

…

19: 28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions

… ►With ${𝒞}_{\mu }^{(j)}$, $c_n^{\nu }(q)$, $A_n^m(q)$, and $B_n^m(q)$ as in §28.23, … ►

28.24.2 $${\epsilon }_s{\mathrm{Mc}}_{2m}^{(j)}(z,h)={(-1)}^m\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{(-1)}^{\mathrm{\ell }}\frac{A_{2\mathrm{\ell }}^{2m}(h^2)}{A_{2s}^{2m}(h^2)}\left(J_{\mathrm{\ell }-s}\left(h{\mathrm{e}}^{-z}\right){𝒞}_{\mathrm{\ell }+s}^{(j)}(h{\mathrm{e}}^z)+J_{\mathrm{\ell }+s}\left(h{\mathrm{e}}^{-z}\right){𝒞}_{\mathrm{\ell }-s}^{(j)}(h{\mathrm{e}}^z)\right),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{e}$: base of natural logarithm, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions, ${\epsilon }_s$: constants and $A_m(q)$: Fourier coefficient

A&S Ref:

20.6.7

Permalink:

http://dlmf.nist.gov/28.24.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.24 and Ch.28

►

28.24.3 $${\mathrm{Mc}}_{2m+1}^{(j)}(z,h)={(-1)}^m\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{(-1)}^{\mathrm{\ell }}\frac{A_{2\mathrm{\ell }+1}^{2m+1}(h^2)}{A_{2s+1}^{2m+1}(h^2)}\left(J_{\mathrm{\ell }-s}\left(h{\mathrm{e}}^{-z}\right){𝒞}_{\mathrm{\ell }+s+1}^{(j)}(h{\mathrm{e}}^z)+J_{\mathrm{\ell }+s+1}\left(h{\mathrm{e}}^{-z}\right){𝒞}_{\mathrm{\ell }-s}^{(j)}(h{\mathrm{e}}^z)\right),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{e}$: base of natural logarithm, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions and $A_m(q)$: Fourier coefficient

A&S Ref:

20.6.8

Permalink:

http://dlmf.nist.gov/28.24.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.24 and Ch.28

►

28.24.4 $${\mathrm{Ms}}_{2m+1}^{(j)}(z,h)={(-1)}^m\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{(-1)}^{\mathrm{\ell }}\frac{B_{2\mathrm{\ell }+1}^{2m+1}(h^2)}{B_{2s+1}^{2m+1}(h^2)}\left(J_{\mathrm{\ell }-s}\left(h{\mathrm{e}}^{-z}\right){𝒞}_{\mathrm{\ell }+s+1}^{(j)}(h{\mathrm{e}}^z)-J_{\mathrm{\ell }+s+1}\left(h{\mathrm{e}}^{-z}\right){𝒞}_{\mathrm{\ell }-s}^{(j)}(h{\mathrm{e}}^z)\right),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{e}$: base of natural logarithm, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions and $B_m(q)$: Fourier coefficient

A&S Ref:

20.6.9

Permalink:

http://dlmf.nist.gov/28.24.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.24 and Ch.28

►

28.24.5 $${\mathrm{Ms}}_{2m+2}^{(j)}(z,h)={(-1)}^m\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{(-1)}^{\mathrm{\ell }}\frac{B_{2\mathrm{\ell }+2}^{2m+2}(h^2)}{B_{2s+2}^{2m+2}(h^2)}\left(J_{\mathrm{\ell }-s}\left(h{\mathrm{e}}^{-z}\right){𝒞}_{\mathrm{\ell }+s+2}^{(j)}(h{\mathrm{e}}^z)-J_{\mathrm{\ell }+s+2}\left(h{\mathrm{e}}^{-z}\right){𝒞}_{\mathrm{\ell }-s}^{(j)}(h{\mathrm{e}}^z)\right),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{e}$: base of natural logarithm, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions and $B_m(q)$: Fourier coefficient

A&S Ref:

20.6.10

Permalink:

http://dlmf.nist.gov/28.24.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.24 and Ch.28

…

20: 2.4 Contour Integrals

… ►Let $𝒫$ denote the path for the contour integral …in which $a$ is finite, $b$ is finite or infinite, and $\omega$ is the angle of slope of $𝒫$ at $a$, that is, $lim(\mathrm{ph}\left(t-a\right))$ as $t\to a$ along $𝒫$. Assume that $p(t)$ and $q(t)$ are analytic on an open domain $𝐓$ that contains $𝒫$, with the possible exceptions of $t=a$ and $t=b$. … ►Now suppose that in (2.4.10) the minimum of $\mathrm{\Re }\left(zp(t)\right)$ on $𝒫$ occurs at an interior point $t_0$. … ►where $𝒬$ is the $w$-map of $𝒫$, and …