Weierstrass zeta function

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11: 23.9 Laurent and Other Power Series

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23.9.3 $$\zeta \left(z\right)=\frac{1}{z}-\sum _{n=2}^{\mathrm{\infty }}\frac{c_n}{2n-1}{z}^{2n-1},$$ .

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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 and $c_n$

A&S Ref:

18.5.5

Referenced by:

§23.9

Permalink:

http://dlmf.nist.gov/23.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §23.9 and Ch.23

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12: 23.12 Asymptotic Approximations

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23.12.2 $$\zeta \left(z\right)=\frac{{\pi }^2}{4{\omega }_1^2}\left(\frac{z}{3}+\frac{2{\omega }_1}{\pi }\cot \left(\frac{\pi z}{2{\omega }_1}\right)-8\left(z-\frac{{\omega }_1}{\pi }\sin \left(\frac{\pi z}{{\omega }_1}\right)\right)q^2+O\left(q^4\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\sin z$: sine function, $𝕃$: lattice, $q$: nome, $z$: complex and ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators

Referenced by:

§23.12, Erratum (V1.0.23) for Equation (23.12.2)

Permalink:

http://dlmf.nist.gov/23.12.E2

Encodings:

pMML, png, TeX

Correction (effective with 1.0.23):

The factor of 2, previously omitted from the denominator of the argument of the $\cot$ function has been inserted.

Suggested 2019-05-27 by Blagoje Oblak

See also:

Annotations for §23.12 and Ch.23

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13: 23.8 Trigonometric Series and Products

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23.8.2 $$\zeta \left(z\right)-\frac{{\eta }_{1}z}{{\omega }_1}-\frac{\pi }{2{\omega }_1}\cot \left(\frac{\pi z}{2{\omega }_1}\right)=\frac{2\pi }{{\omega }_1}\sum _{n=1}^{\mathrm{\infty }}\frac{q^{2n}}{1-q^{2n}}\sin \left(\frac{n\pi z}{{\omega }_1}\right).$$

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Symbols:

$\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\sin z$: sine function, $𝕃$: lattice, $q$: nome, $n$: integer, $z$: complex, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and ${\eta }_j$: complex number

Permalink:

http://dlmf.nist.gov/23.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §23.8(i), §23.8 and Ch.23

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23.8.4 $$\zeta \left(z\right)=\frac{{\eta }_{1}z}{{\omega }_1}+\frac{\pi }{2{\omega }_1}\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\cot \left(\frac{\pi (z+2n{\omega }_3)}{2{\omega }_1}\right),$$

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Symbols:

$\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent 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:

§23.8(ii)

Permalink:

http://dlmf.nist.gov/23.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §23.8(ii), §23.8 and Ch.23

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14: 19.25 Relations to Other Functions

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19.25.37 $$\zeta \left(z+2\omega \right)+(z+2\omega )\mathrm{\wp }\left(z\right)=\pm 2R_G(\mathrm{\wp }\left(z\right)-e_1,\mathrm{\wp }\left(z\right)-e_2,\mathrm{\wp }\left(z\right)-e_3),$$

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Symbols:

$R_G(x,y,z)$: symmetric elliptic integral of second kind, $\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $e_j$: Weierstrass lattice roots, $\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function and $z$: complex number

Referenced by:

§19.25(vi), Erratum (V1.0.18) for Equation (19.25.37), Erratum (V1.1.0) for Subsection 19.25(vi)

Permalink:

http://dlmf.nist.gov/19.25.E37

Encodings:

pMML, png, TeX

Correction (effective with 1.1.0):

This equation has been corrected, namely the left-hand side which was originally $\zeta \left(z\right)+z\mathrm{\wp }\left(z\right)$, has been replaced by $\zeta \left(z+2\omega \right)+(z+2\omega )\mathrm{\wp }\left(z\right)$, and the right-hand side has been multiplied by $\pm 1$.

Clarification (effective with 1.0.18):

The Weierstrasszetafunction in this equation incorrectly linked to the definition of the Riemann zetafunction. However, to the eye, the functions appeared correct.

See also:

Annotations for §19.25(vi), §19.25 and Ch.19

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19.25.39 $$\zeta \left({\omega }_j\right)+{\omega }_j{e}_j=2R_G(0,e_j-e_k,e_j-e_{\mathrm{\ell }}),$$

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Symbols:

$R_G(x,y,z)$: symmetric elliptic integral of second kind, $e_j$: Weierstrass lattice roots, $\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $j$: $(=1,2,3)$ , $z$: complex number, $k$: $(=1,2,3)$ , $\mathrm{\ell }$: $(=1,2,3)$ , ${\omega }_j$: nonzero complex number and ${\eta }_j$: complex number

Proof sketch:

Take $z=-\omega$ in (19.25.36).

Referenced by:

§19.25(vi), §19.25(vi), Erratum (V1.1.0) for Subsection 19.25(vi)

Permalink:

http://dlmf.nist.gov/19.25.E39

Encodings:

pMML, png, TeX

Correction (effective with 1.1.0):

This equation has been corrected such that the left-hand side ${\eta }_j$ has been replaced by $\zeta \left({\omega }_j\right)$.

See also:

Annotations for §19.25(vi), §19.25 and Ch.19

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15: 23.21 Physical Applications

§23.21 Physical Applications

… ►The Weierstrass function $\mathrm{\wp }$ plays a similar role for cubic potentials in canonical form $g_3+g_{2}x-4x^3$. … ►

§23.21(ii) Nonlinear Evolution Equations

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§23.21(iii) Ellipsoidal Coordinates

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String theory. See Green et al. (1988a, §8.2) and Polchinski (1998, §7.2).

16: 22.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) … ►The functions treated in this chapter are the three principal Jacobian elliptic functions $\mathrm{sn}(z,k)$, $\mathrm{cn}(z,k)$, $\mathrm{dn}(z,k)$; the nine subsidiary Jacobian elliptic functions $\mathrm{cd}(z,k)$, $\mathrm{sd}(z,k)$, $\mathrm{nd}(z,k)$, $\mathrm{dc}(z,k)$, $\mathrm{nc}(z,k)$, $\mathrm{sc}(z,k)$, $\mathrm{ns}(z,k)$, $\mathrm{ds}(z,k)$, $\mathrm{cs}(z,k)$; the amplitude function $\mathrm{am}(x,k)$; Jacobi’s epsilon and zeta functions $ℰ(x,k)$ and $\mathrm{Z}\left(x|k\right)$. … ►Similarly for the other functions.

17: Errata

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23.12.2 $$\zeta \left(z\right)=\frac{{\pi }^2}{4{\omega }_1^2}\left(\frac{z}{3}+\frac{2{\omega }_1}{\pi }\cot \left(\frac{\pi z}{2{\omega }_1}\right)-8\left(z-\frac{{\omega }_1}{\pi }\sin \left(\frac{\pi z}{{\omega }_1}\right)\right)q^2+O\left(q^4\right)\right)$$

Originally, the factor of 2 was missing from the denominator of the argument of the $\cot$ function.

Reported by Blagoje Oblak on 2019-05-27

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The Weierstrass zeta function was incorrectly linked to the definition of the Riemann zeta function. However, to the eye, the function appeared correct. The link was corrected.

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18: 23.23 Tables

§23.23 Tables

… ►2 in Abramowitz and Stegun (1964) gives values of $\mathrm{\wp }\left(z\right)$, ${\mathrm{\wp }}^{\prime }\left(z\right)$, and $\zeta \left(z\right)$ to 7 or 8D in the rectangular and rhombic cases, normalized so that ${\omega }_1=1$ and ${\omega }_3=\mathrm{i}a$ (rectangular case), or ${\omega }_1=1$ and ${\omega }_3=\frac{1}{2}+\mathrm{i}a$ (rhombic case), for $a$ = 1. …05, and in the case of $\mathrm{\wp }\left(z\right)$ the user may deduce values for complex $z$ by application of the addition theorem (23.10.1). ►Abramowitz and Stegun (1964) also includes other tables to assist the computation of the Weierstrass functions, for example, the generators as functions of the lattice invariants $g_2$ and $g_3$. ►For earlier tables related to Weierstrass functions see Fletcher et al. (1962, pp. 503–505) and Lebedev and Fedorova (1960, pp. 223–226).

19: 29.2 Differential Equations

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29.2.11 $$\zeta =\mathrm{\wp }(\eta ;g_2,g_3)=\mathrm{\wp }\left(\eta \right),$$

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Defines:

$\zeta$ (locally)

Symbols:

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

Permalink:

http://dlmf.nist.gov/29.2.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §29.2(ii), §29.2 and Ch.29

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20: 31.2 Differential Equations

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Weierstrass’s Form

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$\zeta =\mathrm{i}{K}^{\prime }+\xi {(e_1-e_3)}^{1/2},$

… ►where $2{\omega }_1$ and $2{\omega }_3$ with $\mathrm{\Im }\left({\omega }_3/{\omega }_1\right)>0$ are generators of the lattice $𝕃$ for $\mathrm{\wp }\left(z|𝕃\right)$. … ►

31.2.10 $$w(\xi )={\left(\mathrm{\wp }\left(\xi \right)-e_3\right)}^{(1-2\gamma )/4}{\left(\mathrm{\wp }\left(\xi \right)-e_2\right)}^{(1-2\delta )/4}{\left(\mathrm{\wp }\left(\xi \right)-e_1\right)}^{(1-2ϵ)/4}W(\xi ),$$

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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, $e_j$: Weierstrass lattice roots, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $𝕃$: lattice and $W(\xi )$: solution

Permalink:

http://dlmf.nist.gov/31.2.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §31.2(iv), §31.2(iv), §31.2 and Ch.31

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31.2.11 $$d^{2}W/{d\xi }^2+\left(H+b_0\mathrm{\wp }\left(\xi \right)+b_1\mathrm{\wp }\left(\xi +{\omega }_1\right)+b_2\mathrm{\wp }\left(\xi +{\omega }_2\right)+b_3\mathrm{\wp }\left(\xi +{\omega }_3\right)\right)W=0,$$

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Defines:

$W(\xi )$: solution (locally)

Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , ${\omega }_j$: nonzero complex number, $𝕃$: lattice, $b_j$: coefficients and $H$

Permalink:

http://dlmf.nist.gov/31.2.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §31.2(iv), §31.2(iv), §31.2 and Ch.31

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