Weierstrass zeta function

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1: 23.1 Special Notation

… ►The main functions treated in this chapter are the Weierstrass $\mathrm{\wp }$-function $\mathrm{\wp }\left(z\right)=\mathrm{\wp }\left(z|𝕃\right)=\mathrm{\wp }(z;g_2,g_3)$; the Weierstrass zeta function $\zeta \left(z\right)=\zeta \left(z|𝕃\right)=\zeta (z;g_2,g_3)$; the Weierstrass sigma function $\sigma \left(z\right)=\sigma \left(z|𝕃\right)=\sigma (z;g_2,g_3)$; the elliptic modular function $\lambda \left(\tau \right)$; Klein’s complete invariant $J\left(\tau \right)$; Dedekind’s eta function $\eta \left(\tau \right)$. …

2: 23.14 Integrals

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23.14.1 $$\int \mathrm{\wp }\left(z\right)dz=-\zeta \left(z\right),$$

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Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $dx$: differential of $x$ , $\int$: integral, $𝕃$: 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

… ►

23.14.3 $$\int {\mathrm{\wp }}^3\left(z\right)dz=\frac{1}{120}{\mathrm{\wp }}^{\prime \prime \prime }\left(z\right)-\frac{3}{20}{g}_2\zeta \left(z\right)+\frac{1}{10}{g}_{3}z.$$

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Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $g_j$: Weierstrass lattice invariants $g_2$, $g_3$ , $\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $dx$: differential of $x$ , $\int$: integral, $𝕃$: 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

…

3: 23.2 Definitions and Periodic Properties

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23.2.5 $$\zeta \left(z\right)=\frac{1}{z}+\sum _{w\in 𝕃\setminus \{0\}}\left(\frac{1}{z-w}+\frac{1}{w}+\frac{z}{w^2}\right),$$

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Defines:

$\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function

Symbols:

$g_j$: Weierstrass lattice invariants $g_2$, $g_3$ , $\zeta (z;g_2,g_3)$: Weierstrass zeta function, $\in$: element of, $\setminus$: set subtraction, $𝕃$: lattice and $z$: complex

Keywords:

modular functions, Weierstrasszetafunction

Referenced by:

§23.9

Permalink:

http://dlmf.nist.gov/23.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §23.2(ii), §23.2 and Ch.23

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23.2.7 $$\mathrm{\wp }\left(z\right)=-{\zeta }^{\prime }\left(z\right),$$

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Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $𝕃$: lattice and $z$: complex

A&S Ref:

18.6.1

Referenced by:

§23.14, §23.3(ii)

Permalink:

http://dlmf.nist.gov/23.2.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §23.2(ii), §23.2 and Ch.23

►

23.2.8 $$\zeta \left(z\right)={\sigma }^{\prime }\left(z\right)/\sigma \left(z\right).$$

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Symbols:

$\sigma \left(z\right)$ (= $\sigma \left(z|𝕃\right)$ = $\sigma (z;g_2,g_3)$): Weierstrass sigma function, $\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $𝕃$: lattice and $z$: complex

A&S Ref:

18.6.2

Referenced by:

§23.3(ii), §23.9

Permalink:

http://dlmf.nist.gov/23.2.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §23.2(ii), §23.2 and Ch.23

► $\mathrm{\wp }\left(z\right)$ and $\zeta \left(z\right)$ are meromorphic functions with poles at the lattice points. … ►The function $\zeta \left(z\right)$ is quasi-periodic: for $j=1,2,3$, …

4: 25.1 Special Notation

… ►The main related functions are the Hurwitz zeta function $\zeta (s,a)$, the dilogarithm ${\mathrm{Li}}_2\left(z\right)$, the polylogarithm ${\mathrm{Li}}_s\left(z\right)$ (also known as Jonquière’s function $\varphi (z,s)$), Lerch’s transcendent $\mathrm{\Phi }(z,s,a)$, and the Dirichlet $L$-functions $L(s,\chi )$.

5: 23.3 Differential Equations

… ►As functions of $g_2$ and $g_3$, $\mathrm{\wp }(z;g_2,g_3)$ and $\zeta (z;g_2,g_3)$ are meromorphic and $\sigma (z;g_2,g_3)$ is entire. …

6: 23.4 Graphics

… ►Line graphs of the Weierstrass functions $\mathrm{\wp }\left(x\right)$, $\zeta \left(x\right)$, and $\sigma \left(x\right)$, illustrating the lemniscatic and equianharmonic cases. … ►Surfaces for the Weierstrass functions $\mathrm{\wp }\left(z\right)$, $\zeta \left(z\right)$, and $\sigma \left(z\right)$. …

7: 23.11 Integral Representations

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23.11.3 $$\zeta \left(z\right)=\frac{1}{z}+{\int }_0^{\mathrm{\infty }}\left({\mathrm{e}}^{-s}\left(zs-\sinh \left(zs\right)\right)f_1(s,\tau )-{\mathrm{e}}^{\mathrm{i}\tau s}\left(zs-\sin \left(zs\right)\right)f_2(s,\tau )\right)ds,$$

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Symbols:

$\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $𝕃$: 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: 23.6 Relations to Other Functions

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23.6.13 $$\zeta \left(u\right)=\frac{{\eta }_1}{{\omega }_1}u+\frac{\pi }{2{\omega }_1}\frac{d}{dz}\ln {\theta }_1(z,q),$$

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Symbols:

${\theta }_j(z,q)$: theta function, $\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\ln z$: principal branch of logarithm function, $𝕃$: lattice, $q$: nome, $z$: complex, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators, ${\eta }_j$: complex number and $u$: change of variable

Referenced by:

§23.22(i), §23.6(i)

Permalink:

http://dlmf.nist.gov/23.6.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §23.6(i), §23.6 and Ch.23

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23.6.27 $$\zeta \left(z|{𝕃}_1\right)-\zeta \left(z+2K|{𝕃}_1\right)+\zeta \left(2K|{𝕃}_1\right)=\mathrm{ns}(z,k),$$

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Symbols:

$\mathrm{ns}(z,k)$: Jacobian elliptic function, $\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $𝕃$: lattice, $z$: complex and $k$: modulus

Referenced by:

§23.6(ii), §23.6(ii)

Permalink:

http://dlmf.nist.gov/23.6.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §23.6(ii), §23.6 and Ch.23

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23.6.28 $$\zeta \left(z|{𝕃}_2\right)-\zeta \left(z+2K|{𝕃}_2\right)+\zeta \left(2K|{𝕃}_2\right)=\mathrm{ds}(z,k),$$

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Symbols:

$\mathrm{ds}(z,k)$: Jacobian elliptic function, $\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $𝕃$: lattice, $z$: complex and $k$: modulus

Permalink:

http://dlmf.nist.gov/23.6.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §23.6(ii), §23.6 and Ch.23

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23.6.29 $$\zeta \left(z|{𝕃}_3\right)-\zeta \left(z+2\mathrm{i}{K}^{\prime }|{𝕃}_3\right)-\zeta \left(2\mathrm{i}{K}^{\prime }|{𝕃}_3\right)=\mathrm{cs}(z,k).$$

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Symbols:

$\mathrm{cs}(z,k)$: Jacobian elliptic function, $\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $𝕃$: lattice, $z$: complex and $k$: modulus

Referenced by:

§23.6(ii), §23.6(ii)

Permalink:

http://dlmf.nist.gov/23.6.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §23.6(ii), §23.6 and Ch.23

… ►For representations of general elliptic functions (§23.2(iii)) in terms of $\sigma \left(z\right)$ and $\mathrm{\wp }\left(z\right)$ see Lawden (1989, §§8.9, 8.10), and for expansions in terms of $\zeta \left(z\right)$ see Lawden (1989, §8.11). …

9: 23.10 Addition Theorems and Other Identities

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23.10.2 $$\zeta \left(u+v\right)=\zeta \left(u\right)+\zeta \left(v\right)+\frac{1}{2}\frac{{\zeta }^{\prime \prime }\left(u\right)-{\zeta }^{\prime \prime }\left(v\right)}{{\zeta }^{\prime }\left(u\right)-{\zeta }^{\prime }\left(v\right)},$$

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Symbols:

$\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function and $𝕃$: lattice

A&S Ref:

18.4.3

Referenced by:

§23.9

Permalink:

http://dlmf.nist.gov/23.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §23.10(i), §23.10 and Ch.23

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23.10.6 $${\left(\zeta \left(u\right)+\zeta \left(v\right)+\zeta \left(w\right)\right)}^2+{\zeta }^{\prime }\left(u\right)+{\zeta }^{\prime }\left(v\right)+{\zeta }^{\prime }\left(w\right)=0.$$

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Symbols:

$\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $𝕃$: lattice and $w=-u-v$

Permalink:

http://dlmf.nist.gov/23.10.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §23.10(i), §23.10 and Ch.23

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23.10.9 $$\zeta \left(2z\right)=2\zeta \left(z\right)+\frac{1}{2}\frac{{\zeta }^{\prime \prime \prime }\left(z\right)}{{\zeta }^{\prime \prime }\left(z\right)},$$

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Symbols:

$\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $𝕃$: lattice and $z$: complex

A&S Ref:

18.4.7

Referenced by:

§23.10(ii)

Permalink:

http://dlmf.nist.gov/23.10.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §23.10(ii), §23.10 and Ch.23

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23.10.12 $$n\zeta \left(nz\right)=-n(n-1)({\eta }_1+{\eta }_3)+\sum _{j=0}^{n-1}\sum _{\mathrm{\ell }=0}^{n-1}\zeta \left(z+\frac{2j}{n}{\omega }_1+\frac{2\mathrm{\ell }}{n}{\omega }_3\right),$$

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Symbols:

$\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $𝕃$: lattice, $n$: integer, $z$: complex, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and ${\eta }_j$: complex number

Referenced by:

§23.10(iii)

Permalink:

http://dlmf.nist.gov/23.10.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §23.10(iii), §23.10 and Ch.23

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23.10.18 $$\zeta \left(cz|c𝕃\right)=c^{-1}\zeta \left(z|𝕃\right),$$

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Symbols:

$\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $𝕃$: 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

…

10: 23.22 Methods of Computation

… ►The functions $\zeta \left(z\right)$ and $\sigma \left(z\right)$ are computed in a similar manner: the former by replacing $u$ and $z$ in (23.6.13) by $z$ and $\pi z/(2{\omega }_1)$, respectively, and also referring to (23.6.8); the latter by applying (23.6.9). …