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1: 23.10 Addition Theorems and Other Identities

… ►(23.10.8) continues to hold when

e_{1}

e_{2}

e_{3}

are permuted cyclically. … ►where … ►

23.10.17

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

ⓘ

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

►

23.10.18

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

ⓘ

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

… ►Also, 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. …

2: 23.9 Laurent and Other Power Series

… ►Let

z_{0}(\neq 0)

be the nearest lattice point to the origin, and define …Explicit coefficients

c_{n}

in terms of

c_{2}

and

c_{3}

are given up to

c_{19}

in Abramowitz and Stegun (1964, p. 636). ►For

j=1,2,3

, and with

e_{j}

as in §23.3(i), … ►where

a_{0,0}=1

a_{m,n}=0

if either

m

n<0

, and …For

a_{m,n}

with

m=0,1,\dots,12

and

n=0,1,\dots,8

, see Abramowitz and Stegun (1964, p. 637).

3: 23.14 Integrals

… ►

23.14.1

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

ⓘ

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

►

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,

ⓘ

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

►

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.

ⓘ

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

…

4: 23.2 Definitions and Periodic Properties

… ►The 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 … ►When

z\notin\mathbb{L}

the functions are related by … ►When 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. … ►If

2\omega_{1}

2\omega_{3}

is any pair of generators of

\mathbb{L}

, and

\omega_{2}

is defined by (23.2.1), then …

5: 23.12 Asymptotic Approximations

… ►If

q\>(=e^{\pi i\omega_{3}/\omega_{1}})\to 0

with

\omega_{1}

and

z

fixed, then ►

23.12.1

\wp\left(z\right)=\frac{\pi^{2}}{4\omega_{1}^{2}}\left(-\frac{1}{3}+{\csc}^{2}% \left(\frac{\pi z}{2\omega_{1}}\right)+8\left(1-\cos\left(\frac{\pi z}{\omega_% {1}}\right)\right)q^{2}+O\left(q^{4}\right)\right),

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\csc\NVar{z}$ : cosecant function, $\cos\NVar{z}$ : cosine function, $\mathbb{L}$ : lattice, $q$ : nome, $z$ : complex and $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators
Referenced by:: §23.12
Permalink:: http://dlmf.nist.gov/23.12.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §23.12 and Ch.23

… ►provided that

z\notin\mathbb{L}

in the case of (23.12.1) and (23.12.2). Also, …with similar results for

\eta_{2}

and

\eta_{3}

obtainable by use of (23.2.14).

6: 31.2 Differential Equations

… ►where

2\omega_{1}

and

2\omega_{3}

with

\Im\left(\omega_{3}/\omega_{1}\right)>0

are generators of the lattice

\mathbb{L}

for

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

. … ►

w(z)=z^{1-\gamma}w_{1}(z)

satisfies (31.2.1) if

w_{1}

is a solution of (31.2.1) with transformed parameters

q_{1}=q+(a\delta+\epsilon)(1-\gamma)

;

\alpha_{1}=\alpha+1-\gamma

\beta_{1}=\beta+1-\gamma

\gamma_{1}=2-\gamma

. …By composing these three steps, there result

2^{3}=8

possible transformations of the dependent variable (including the identity transformation) that preserve the form of (31.2.1). … ►There are

4!=24

homographies

\tilde{z}(z)=(Az+B)/(Cz+D)

that take

0,1,a,\infty

to some permutation of

0,1,a^{\prime},\infty

, where

a^{\prime}

may differ from

a

. … ►There are

8\cdot 24=192

automorphisms of equation (31.2.1) by compositions of

F

-homotopic and homographic transformations. …

7: 23.11 Integral Representations

… ►Let

\tau=\ifrac{\omega_{3}}{\omega_{1}}

and ►

f_{1}(s,\tau)=\frac{{\cosh}^{2}\left(\tfrac{1}{2}\tau s\right)}{1-2e^{-s}\cosh% \left(\tau s\right)+e^{-2s}},

►

f_{2}(s,\tau)=\frac{{\cos}^{2}\left(\tfrac{1}{2}s\right)}{1-2e^{i\tau s}\cos s% +e^{2i\tau s}}.

… ►

23.11.2

\wp\left(z\right)=\frac{1}{z^{2}}+8\int_{0}^{\infty}s\left(e^{-s}{\sinh}^{2}% \left(\tfrac{1}{2}zs\right)f_{1}(s,\tau)+e^{i\tau s}{\sin}^{2}\left(\tfrac{1}{% 2}zs\right)f_{2}(s,\tau)\right)\,\mathrm{d}s,

ⓘ

Symbols:: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\sin\NVar{z}$ : sine function, $\mathbb{L}$ : lattice, $z$ : complex, $\tau$ : complex variable and $f_{j}(s,\tau)$ : functions
Permalink:: http://dlmf.nist.gov/23.11.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §23.11 and Ch.23

… ►

23.11.3

\zeta\left(z\right)=\frac{1}{z}+\int_{0}^{\infty}\left(e^{-s}\left(zs-\sinh% \left(zs\right)\right)f_{1}(s,\tau)-e^{i\tau s}\left(zs-\sin\left(zs\right)% \right)f_{2}(s,\tau)\right)\,\mathrm{d}s,

ⓘ

Symbols:: $\zeta\left(\NVar{z}\right)$ (= $\zeta\left(z|\mathbb{L}\right)$ = $\zeta\left(z;g_{2},g_{3}\right)$ ): Weierstrass zeta function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\sin\NVar{z}$ : sine function, $\mathbb{L}$ : lattice, $z$ : complex, $\tau$ : complex variable and $f_{j}(s,\tau)$ : functions
Permalink:: http://dlmf.nist.gov/23.11.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §23.11 and Ch.23

…

8: 23.7 Quarter Periods

… ►

23.7.1

\wp\left(\tfrac{1}{2}\omega_{1}\right)=e_{1}+\sqrt{(e_{1}-e_{3})(e_{1}-e_{2})}% =e_{1}+\omega_{1}^{-2}(K\left(k\right))^{2}k^{\prime},

ⓘ

Symbols:: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $e_{\NVar{j}}$ : Weierstrass lattice roots, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathbb{L}$ : lattice, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators, $k$ : modulus and $k^{\prime}$ : complementary modulus
A&S Ref:: 18.3.18
Permalink:: http://dlmf.nist.gov/23.7.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §23.7 and Ch.23

►

23.7.2

\wp\left(\tfrac{1}{2}\omega_{2}\right)=e_{2}-i\sqrt{(e_{1}-e_{2})(e_{2}-e_{3})% }=e_{2}-i\omega_{1}^{-2}(K\left(k\right))^{2}kk^{\prime},

ⓘ

Symbols:: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $e_{\NVar{j}}$ : Weierstrass lattice roots, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$ : imaginary unit, $\mathbb{L}$ : lattice, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators, $k$ : modulus and $k^{\prime}$ : complementary modulus
A&S Ref:: 18.3.18
Permalink:: http://dlmf.nist.gov/23.7.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §23.7 and Ch.23

►

23.7.3

\wp\left(\tfrac{1}{2}\omega_{3}\right)=e_{3}-\sqrt{(e_{1}-e_{3})(e_{2}-e_{3})}% =e_{3}-\omega_{1}^{-2}(K\left(k\right))^{2}k,

ⓘ

Symbols:: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $e_{\NVar{j}}$ : Weierstrass lattice roots, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathbb{L}$ : lattice, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $k$ : modulus
A&S Ref:: 18.3.18
Permalink:: http://dlmf.nist.gov/23.7.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §23.7 and Ch.23

…

9: 23.19 Interrelations

… ►

23.19.1

\lambda\left(\tau\right)=16\left(\frac{{\eta}^{2}\left(2\tau\right)\eta\left(% \tfrac{1}{2}\tau\right)}{{\eta}^{3}\left(\tau\right)}\right)^{8},

ⓘ

Symbols:: $\eta\left(\NVar{\tau}\right)$ : Dedekind’s eta function (or Dedekind modular function), $\lambda\left(\NVar{\tau}\right)$ : elliptic modular function and $\tau$ : complex variable
Permalink:: http://dlmf.nist.gov/23.19.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §23.19 and Ch.23

… ►

23.19.3

J\left(\tau\right)=\frac{{g_{2}}^{3}}{{g_{2}}^{3}-27{g_{3}}^{2}},

ⓘ

Symbols:: $J\left(\NVar{\tau}\right)$ : Klein’s complete invariant, $g_{\NVar{j}}$ : Weierstrass lattice invariants $g_{2}$ , $g_{3}$ , $\mathbb{L}$ : lattice and $\tau$ : complex variable
Permalink:: http://dlmf.nist.gov/23.19.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §23.19 and Ch.23

►where

g_{2},g_{3}

are the invariants of the lattice

\mathbb{L}

with generators

1

and

\tau

; see §23.3(i). …

10: 23.3 Differential Equations

… ►and are denoted by

e_{1},e_{2},e_{3}

. … ►Let

{g_{2}}^{3}\neq 27{g_{3}}^{2}

, or equivalently

\Delta

be nonzero, or

e_{1},e_{2},e_{3}

be distinct. Given

g_{2}

and

g_{3}

there is a unique lattice

\mathbb{L}

such that (23.3.1) and (23.3.2) are satisfied. … ►Conversely,

g_{2}

g_{3}

, and the set

\{e_{1},e_{2},e_{3}\}

are determined uniquely by the lattice

\mathbb{L}

independently of the choice of generators. However, given any pair of generators

2\omega_{1}

2\omega_{3}

\mathbb{L}

, and with

\omega_{2}

defined by (23.2.1), we can identify the

e_{j}

individually, via …