# Weierstrass zeta function

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## 1—10 of 23 matching pages

##### 1: 23.1 Special Notation
βΊThe main functions treated in this chapter are the Weierstrass $\wp$-function $\wp\left(z\right)=\wp\left(z|\mathbb{L}\right)=\wp\left(z;g_{2},g_{3}\right)$; the Weierstrass zeta function $\zeta\left(z\right)=\zeta\left(z|\mathbb{L}\right)=\zeta\left(z;g_{2},g_{3}\right)$; the Weierstrass sigma function $\sigma\left(z\right)=\sigma\left(z|\mathbb{L}\right)=\sigma\left(z;g_{2},g_{3}\right)$; 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 $\int\wp\left(z\right)\,\mathrm{d}z=-\zeta\left(z\right),$
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23.14.3 $\int{\wp}^{3}\left(z\right)\,\mathrm{d}z=\frac{1}{120}\wp'''\left(z\right)-% \frac{3}{20}g_{2}\zeta\left(z\right)+\frac{1}{10}g_{3}z.$
##### 3: 23.2 Definitions and Periodic Properties
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23.2.5 $\zeta\left(z\right)=\frac{1}{z}+\sum_{w\in\mathbb{L}\setminus\{0\}}\left(\frac% {1}{z-w}+\frac{1}{w}+\frac{z}{w^{2}}\right),$
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23.2.7 $\wp\left(z\right)=-\zeta'\left(z\right),$
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23.2.8 $\zeta\left(z\right)=\ifrac{\sigma'\left(z\right)}{\sigma\left(z\right)}.$
βΊ $\wp\left(z\right)$ and $\zeta\left(z\right)$ are meromorphic functions with poles at the lattice points. … βΊThe function $\zeta\left(z\right)$ is quasi-periodic: for $j=1,2,3$, …
##### 4: 25.1 Special Notation
βΊThe main related functions are the Hurwitz zeta function $\zeta\left(s,a\right)$, the dilogarithm $\operatorname{Li}_{2}\left(z\right)$, the polylogarithm $\operatorname{Li}_{s}\left(z\right)$ (also known as Jonquière’s function $\phi\left(z,s\right)$), Lerch’s transcendent $\Phi\left(z,s,a\right)$, and the Dirichlet $L$-functions $L\left(s,\chi\right)$.
##### 5: 23.3 Differential Equations
βΊAs functions of $g_{2}$ and $g_{3}$, $\wp\left(z;g_{2},g_{3}\right)$ and $\zeta\left(z;g_{2},g_{3}\right)$ are meromorphic and $\sigma\left(z;g_{2},g_{3}\right)$ is entire. …
##### 6: 23.4 Graphics
βΊLine graphs of the Weierstrass functions $\wp\left(x\right)$, $\zeta\left(x\right)$, and $\sigma\left(x\right)$, illustrating the lemniscatic and equianharmonic cases. … βΊSurfaces for the Weierstrass functions $\wp\left(z\right)$, $\zeta\left(z\right)$, and $\sigma\left(z\right)$. …
##### 7: 23.11 Integral Representations
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23.11.3 $\zeta\left(z\right)=\frac{1}{z}+\int_{0}^{\infty}\left(e^{-s}\left(zs-\sinh% \left(zs\right)\right)f_{1}(s,\tau)-e^{i\tau s}\left(zs-\sin\left(zs\right)% \right)f_{2}(s,\tau)\right)\,\mathrm{d}s,$
##### 8: 23.6 Relations to Other Functions
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23.6.13 $\zeta\left(u\right)=\frac{\eta_{1}}{\omega_{1}}u+\frac{\pi}{2\omega_{1}}\frac{% \mathrm{d}}{\mathrm{d}z}\ln\theta_{1}\left(z,q\right),$
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23.6.27 $\zeta\left(z|\mathbb{L}_{\mspace{1.0mu}1}\right)-\zeta\left(z+2K|\mathbb{L}_{% \mspace{1.0mu}1}\right)+\zeta\left(2K|\mathbb{L}_{\mspace{1.0mu}1}\right)=% \operatorname{ns}\left(z,k\right),$
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23.6.28 $\zeta\left(z|\mathbb{L}_{\mspace{1.0mu}2}\right)-\zeta\left(z+2K|\mathbb{L}_{% \mspace{1.0mu}2}\right)+\zeta\left(2K|\mathbb{L}_{\mspace{1.0mu}2}\right)=% \operatorname{ds}\left(z,k\right),$
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23.6.29 $\zeta\left(z|\mathbb{L}_{\mspace{1.0mu}3}\right)-\zeta\left(z+2\mathrm{i}{K^{% \prime}}|\mathbb{L}_{\mspace{1.0mu}3}\right)-\zeta\left(2\mathrm{i}{K^{\prime}% }|\mathbb{L}_{\mspace{1.0mu}3}\right)=\operatorname{cs}\left(z,k\right).$
βΊFor representations of general elliptic functions23.2(iii)) in terms of $\sigma\left(z\right)$ and $\wp\left(z\right)$ 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 $\zeta\left(u+v\right)=\zeta\left(u\right)+\zeta\left(v\right)+\frac{1}{2}\frac% {\zeta''\left(u\right)-\zeta''\left(v\right)}{\zeta'\left(u\right)-\zeta'\left% (v\right)},$
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23.10.6 $\left(\zeta\left(u\right)+\zeta\left(v\right)+\zeta\left(w\right)\right)^{2}+% \zeta'\left(u\right)+\zeta'\left(v\right)+\zeta'\left(w\right)=0.$
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23.10.12 $n\zeta\left(nz\right)=-n(n-1)(\eta_{1}+\eta_{3})+\sum_{j=0}^{n-1}\sum_{\ell=0}% ^{n-1}\zeta\left(z+\frac{2j}{n}\omega_{1}+\frac{2\ell}{n}\omega_{3}\right),$
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##### 10: 23.22 Methods of Computation
βΊThe 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). …