Weierstrass%20zeta%20function

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1: 23.14 Integrals

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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

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2: 23.9 Laurent and Other Power Series

§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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$c_2={\displaystyle \frac{1}{20}}{g}_2,$

… ►For $j=1,2,3$, and with $e_j$ as in §23.3(i), …Also, Abramowitz and Stegun (1964, (18.5.25)) supplies the first 22 terms in the reverted form of (23.9.2) as $1/\mathrm{\wp }\left(z\right)\to 0$. …

3: Software Index

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	Open Source													With Book						Commercial
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20 Theta Functions
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►‘✓’ indicates that a software package implements the functions in a section; ‘a’ indicates available functionality through optional or add-on packages; an empty space indicates no known support. … ►In the list below we identify four main sources of software for computing special functions. … ►

Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

… ►The following are web-based software repositories with significant holdings in the area of special functions. …

4: Errata

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Specific source citations and proof metadata are now given for all equations in Chapter 25 Zeta and Related Functions.

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This subsection has been significantly updated. In particular, the following formulae have been corrected. Equation (19.25.35) has been replaced by

19.25.35 $$z+2\omega =\pm R_F(\mathrm{\wp }\left(z\right)-e_1,\mathrm{\wp }\left(z\right)-e_2,\mathrm{\wp }\left(z\right)-e_3),$$

in which the left-hand side $z$ has been replaced by $z+2\omega$ for some $2\omega \in 𝕃$, and the right-hand side has been multiplied by $\pm 1$. Equation (19.25.37) has been replaced by

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

in which the left-hand side $\zeta \left(z\right)+z\mathrm{\wp }\left(z\right)$ 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 $\pm 1$. Equation (19.25.39) has been replaced by

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

in which the left-hand side ${\eta }_j$ was replaced by $\zeta \left({\omega }_j\right)$, for some $2{\omega }_j\in 𝕃$ and $\mathrm{\wp }\left({\omega }_j\right)=e_j$. Equation (19.25.40) has been replaced by

19.25.40 $$z+2\omega =\pm \sigma \left(z\right)R_F({\sigma }_1^2(z),{\sigma }_2^2(z),{\sigma }_3^2(z)),$$

in which the left-hand side $z$ has been replaced by $z+2\omega$, and the right-hand side was multiplied by $\pm 1$. For more details see §19.25(vi).

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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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Several new equations have been added. See (8.17.24), (20.7.34), §20.11(v), (26.12.27), (36.2.28), and (36.2.29).

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