Weierstrass sigma function

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11: 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). …

12: William P. Reinhardt

… ►Reinhardt firmly believes that the Mandelbrot set is a special function, and notes with interest that the natural boundaries of analyticity of many “more normal” special functions are also fractals. … ►He has been a National Lecturer for Sigma Xi and Phi Beta Kappa, as well as a Sloan, Dreyfus, and Guggenheim Fellow, and Fulbright Senior Scholar (Australia). … ►

20 Theta Functions

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22 Jacobian Elliptic Functions

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23 Weierstrass Elliptic and Modular Functions

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13: 25.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) ► ►►►

$k,m,n$	nonnegative integers.
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$s=\sigma +\mathrm{i}t$	complex variable.
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►The main function treated in this chapter is the Riemann zeta function $\zeta \left(s\right)$. … ►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 $L(s,\chi )$.

14: Errata

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23.6.11 $$\sigma \left({\omega }_2\right)=-2{\omega }_1\frac{\exp \left(\frac{1}{2}{\eta }_1({\omega }_1{\tau }^2+{\omega }_3-{\omega }_2)\right){\theta }_3(0,q)}{\pi q^{1/4}{\theta }_1^{\prime }(0,q)}$$

The factor $2{\omega }_1\mathrm{i}\exp \left(\frac{1}{2}{\eta }_1{\omega }_1{\tau }^2\right)$ has been corrected to be $-2{\omega }_1\exp \left(\frac{1}{2}{\eta }_1({\omega }_1{\tau }^2+{\omega }_3-{\omega }_2)\right)$.

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23.6.12 $$\sigma \left({\omega }_3\right)=2\mathrm{i}{\omega }_1\frac{\exp \left(\frac{1}{2}{\eta }_1{\omega }_1{\tau }^2\right){\theta }_4(0,q)}{\pi q^{1/4}{\theta }_1^{\prime }(0,q)}$$

The factor $-2{\omega }_1\exp \left(\frac{1}{2}{\eta }_1{\omega }_1\right)$ has been corrected to be $2\mathrm{i}{\omega }_1\exp \left(\frac{1}{2}{\eta }_1{\omega }_1{\tau }^2\right)$.

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23.6.15 $$\frac{\sigma \left(u+{\omega }_j\right)}{\sigma \left({\omega }_j\right)}=\exp \left({\eta }_{j}u+\frac{{\eta }_1{u}^2}{2{\omega }_1}\right)\frac{{\theta }_{j+1}(z,q)}{{\theta }_{j+1}(0,q)},$$ $j=1,2,3$

The factor $\exp \left({\eta }_{j}u+\frac{{\eta }_j{u}^2}{2{\omega }_1}\right)$ has been corrected to be $\exp \left({\eta }_{j}u+\frac{{\eta }_1{u}^2}{2{\omega }_1}\right)$.

Reported by Jan Felipe van Diejen on 2021-02-10

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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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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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