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Peter L. Walker

… â–ºPeter L. Walker (b. …

… â–ºShe was one of the LaTeX editors for the DLMF project.

23.10.10

\sigma\left(2z\right)=-\wp'\left(z\right){\sigma}^{4}\left(z\right).

â“˜

Symbols:: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $\sigma\left(\NVar{z}\right)$ (= $\sigma\left(z|\mathbb{L}\right)$ = $\sigma\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $\mathbb{L}$ : lattice and $z$ : complex
A&S Ref:: 18.4.8
Referenced by:: §23.10(ii)
Permalink:: http://dlmf.nist.gov/23.10.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §23.10(ii), §23.10 and Ch.23

… â–º

23.10.17

\wp\left(cz|c\mathbb{L}\right)=c^{-2}\wp\left(z|\mathbb{L}\right),

â“˜

Symbols:: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $\mathbb{L}$ : lattice, $z$ : complex and $c$ : constant
A&S Ref:: 18.2.2
Permalink:: http://dlmf.nist.gov/23.10.E17
Encodings:: pMML, png, TeX
See also:: Annotations for §23.10(iv), §23.10 and Ch.23

â–º

23.10.18

\zeta\left(cz|c\mathbb{L}\right)=c^{-1}\zeta\left(z|\mathbb{L}\right),

â“˜

Symbols:: $\zeta\left(\NVar{z}\right)$ (= $\zeta\left(z|\mathbb{L}\right)$ = $\zeta\left(z;g_{2},g_{3}\right)$ ): Weierstrass zeta function, $\mathbb{L}$ : lattice, $z$ : complex and $c$ : constant
A&S Ref:: 18.2.3
Permalink:: http://dlmf.nist.gov/23.10.E18
Encodings:: pMML, png, TeX
See also:: Annotations for §23.10(iv), §23.10 and Ch.23

â–º

23.10.19

\sigma\left(cz|c\mathbb{L}\right)=c\sigma\left(z|\mathbb{L}\right).

â“˜

Symbols:: $\sigma\left(\NVar{z}\right)$ (= $\sigma\left(z|\mathbb{L}\right)$ = $\sigma\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $\mathbb{L}$ : lattice, $z$ : complex and $c$ : constant
A&S Ref:: 18.2.4
Permalink:: http://dlmf.nist.gov/23.10.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §23.10(iv), §23.10 and Ch.23

â–ºAlso, when

\mathbb{L}

is replaced by

c\mathbb{L}

the lattice invariants

g_{2}

and

g_{3}

are divided by

c^{4}

and

c^{6}

, respectively. …

…

… â–º

23.14.1

\int\wp\left(z\right)\,\mathrm{d}z=-\zeta\left(z\right),

â“˜

Symbols:: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $\zeta\left(\NVar{z}\right)$ (= $\zeta\left(z|\mathbb{L}\right)$ = $\zeta\left(z;g_{2},g_{3}\right)$ ): Weierstrass zeta function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\mathbb{L}$ : lattice and $z$ : complex
A&S Ref:: 18.6.1
Permalink:: http://dlmf.nist.gov/23.14.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §23.14 and Ch.23

â–º

23.14.2

\int{\wp}^{2}\left(z\right)\,\mathrm{d}z=\frac{1}{6}\wp'\left(z\right)+\frac{1% }{12}g_{2}z,

â“˜

Symbols:: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $g_{\NVar{j}}$ : Weierstrass lattice invariants $g_{2}$ , $g_{3}$ , $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\mathbb{L}$ : lattice and $z$ : complex
A&S Ref:: 18.7.1
Permalink:: http://dlmf.nist.gov/23.14.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §23.14 and Ch.23

â–º

23.14.3

\int{\wp}^{3}\left(z\right)\,\mathrm{d}z=\frac{1}{120}\wp'''\left(z\right)-% \frac{3}{20}g_{2}\zeta\left(z\right)+\frac{1}{10}g_{3}z.

â“˜

Symbols:: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $g_{\NVar{j}}$ : Weierstrass lattice invariants $g_{2}$ , $g_{3}$ , $\zeta\left(\NVar{z}\right)$ (= $\zeta\left(z|\mathbb{L}\right)$ = $\zeta\left(z;g_{2},g_{3}\right)$ ): Weierstrass zeta function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\mathbb{L}$ : lattice and $z$ : complex
A&S Ref:: 18.7.2
Permalink:: http://dlmf.nist.gov/23.14.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §23.14 and Ch.23

…

… â–º â–º â–º

$2j_{1},2j_{2},2j_{3},2l_{1},2l_{2},2l_{3}$	nonnegative integers.
…

… â–º

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix},

…

… â–ºThe generators of a given lattice

\mathbb{L}

are not unique. …where

a, b, c, d

are integers, then

2\chi_{1}

,

2\chi_{3}

are generators of

\mathbb{L}

iff … â–ºWhen

z\notin\mathbb{L}

the functions are related by … â–ºWhen it is important to display the lattice with the functions they are denoted by

\wp\left(z|\mathbb{L}\right)

,

\zeta\left(z|\mathbb{L}\right)

, and

\sigma\left(z|\mathbb{L}\right)

, respectively. … â–ºIf

2\omega_{1}

,

2\omega_{3}

is any pair of generators of

\mathbb{L}

, and

\omega_{2}

is defined by (23.2.1), then …

… â–º

See accompanying text — Figure 18.4.5: Laguerre polynomials $L_{n}\left(x\right)$ , $n=1,2,3,4,5$ . Magnify

â–º

… â–º

â–º

… â–ºFor the polynomials

P_{l}

see §18.3, and for the function

Y_{{l},{m}}

see §14.30. â–º

34.3.19

P_{l_{1}}\left(\cos\theta\right)P_{l_{2}}\left(\cos\theta\right)=\sum_{l}(2l+1% )\begin{pmatrix}l_{1}&l_{2}&l\\ 0&0&0\end{pmatrix}^{2}P_{l}\left(\cos\theta\right),

â“˜

Symbols:: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\cos\NVar{z}$ : cosine function, $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol and $l,l_{r}$ : non-negative integers or non-negative integers plus one half.
Referenced by:: §14.18(iv), §14.30(iii), §34.3(vii)
Permalink:: http://dlmf.nist.gov/34.3.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §34.3(vii), §34.3 and Ch.34

â–º

34.3.20

Y_{{l_{1}},{m_{1}}}\left(\theta,\phi\right)Y_{{l_{2}},{m_{2}}}\left(\theta,% \phi\right)=\sum_{l,m}\left(\frac{(2l_{1}+1)(2l_{2}+1)(2l+1)}{4\pi}\right)^{% \frac{1}{2}}\begin{pmatrix}l_{1}&l_{2}&l\\ m_{1}&m_{2}&m\end{pmatrix}\overline{Y_{{l},{m}}\left(\theta,\phi\right)}\begin% {pmatrix}l_{1}&l_{2}&l\\ 0&0&0\end{pmatrix},

â“˜

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\overline{\NVar{z}}$ : complex conjugate, $Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},\NVar{\phi}\right)$ : spherical harmonic, $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol and $l,l_{r}$ : non-negative integers or non-negative integers plus one half.
Permalink:: http://dlmf.nist.gov/34.3.E20
Encodings:: pMML, png, TeX
Notational Change (effective with 1.0.19):: As a notational clarification, the spherical harmonic in the sum over $l, m$ has been explicity marked up as a complex conjugate.
See also:: Annotations for §34.3(vii), §34.3 and Ch.34

â–º

34.3.21

\int_{0}^{\pi}P_{l_{1}}\left(\cos\theta\right)P_{l_{2}}\left(\cos\theta\right)% P_{l_{3}}\left(\cos\theta\right)\sin\theta\,\mathrm{d}\theta=2\begin{pmatrix}l% _{1}&l_{2}&l_{3}\\ 0&0&0\end{pmatrix}^{2},

â“˜

Symbols:: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol and $l,l_{r}$ : non-negative integers or non-negative integers plus one half.
Referenced by:: §14.17(iv)
Permalink:: http://dlmf.nist.gov/34.3.E21
Encodings:: pMML, png, TeX
See also:: Annotations for §34.3(vii), §34.3 and Ch.34

â–º

34.3.22

\int_{0}^{2\pi}\!\int_{0}^{\pi}Y_{{l_{1}},{m_{1}}}\left(\theta,\phi\right)Y_{{% l_{2}},{m_{2}}}\left(\theta,\phi\right)Y_{{l_{3}},{m_{3}}}\left(\theta,\phi% \right)\sin\theta\,\mathrm{d}\theta\,\mathrm{d}\phi=\left(\frac{(2l_{1}+1)(2l_% {2}+1)(2l_{3}+1)}{4\pi}\right)^{\frac{1}{2}}\begin{pmatrix}l_{1}&l_{2}&l_{3}\\ 0&0&0\end{pmatrix}\begin{pmatrix}l_{1}&l_{2}&l_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}.

â“˜

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},\NVar{\phi}\right)$ : spherical harmonic, $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol and $l,l_{r}$ : non-negative integers or non-negative integers plus one half.
Referenced by:: §14.30(ii), §34.3(vii)
Permalink:: http://dlmf.nist.gov/34.3.E22
Encodings:: pMML, png, TeX
See also:: Annotations for §34.3(vii), §34.3 and Ch.34

…

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11—20 of 168 matching pages

11: Peter L. Walker

Profile
Peter L. Walker

12: Gloria Wiersma

13: 23.10 Addition Theorems and Other Identities

14: 10 Bessel Functions

15: 23.14 Integrals

16: 23 Weierstrass Elliptic and Modular
Functions

17: 34.1 Special Notation

18: 23.2 Definitions and Periodic Properties

19: 18.4 Graphics

20: 34.3 Basic Properties: $\mathit{3j}$ Symbol

åŠžå‡çš„ç§‘å¨ç‰¹é©¾é©¶è¯ã€somewhatå¾®aptao168ã€‘ell

11—20 of 168 matching pages

11: Peter L. Walker

Profile Peter L. Walker

12: Gloria Wiersma

13: 23.10 Addition Theorems and Other Identities

14: 10 Bessel Functions

15: 23.14 Integrals

16: 23 Weierstrass Elliptic and Modular Functions

17: 34.1 Special Notation

18: 23.2 Definitions and Periodic Properties

19: 18.4 Graphics

20: 34.3 Basic Properties: 3 â¢ j Symbol

åŠžå‡çš„ç§‘å¨ç‰¹é©¾é©¶è¯ã€somewhatå¾®aptao168ã€‘ell

Profile
Peter L. Walker

16: 23 Weierstrass Elliptic and Modular
Functions

20: 34.3 Basic Properties: $\mathit{3j}$ Symbol