# Weierstrass M-test

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## 1—10 of 36 matching pages

##### 2: 23.3 Differential Equations
The lattice invariants are defined by … and are denoted by $e_{1},e_{2},e_{3}$. … Similarly for $\zeta\left(z;g_{2},g_{3}\right)$ and $\sigma\left(z;g_{2},g_{3}\right)$. As 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. …
##### 3: 23.14 Integrals
###### §23.14 Integrals
23.14.1 $\int\wp\left(z\right)\,\mathrm{d}z=-\zeta\left(z\right),$
23.14.2 $\int{\wp}^{2}\left(z\right)\,\mathrm{d}z=\frac{1}{6}\wp'\left(z\right)+\frac{1% }{12}g_{2}z,$
23.14.3 $\int{\wp}^{3}\left(z\right)\,\mathrm{d}z=\frac{1}{120}\wp'''\left(z\right)-% \frac{3}{20}g_{2}\zeta\left(z\right)+\frac{1}{10}g_{3}z.$
##### 4: 23.7 Quarter Periods
###### §23.7 Quarter Periods
23.7.1 $\wp\left(\tfrac{1}{2}\omega_{1}\right)=e_{1}+\sqrt{(e_{1}-e_{3})(e_{1}-e_{2})}% =e_{1}+\omega_{1}^{-2}(K\left(k\right))^{2}k^{\prime},$
23.7.2 $\wp\left(\tfrac{1}{2}\omega_{2}\right)=e_{2}-i\sqrt{(e_{1}-e_{2})(e_{2}-e_{3})% }=e_{2}-i\omega_{1}^{-2}(K\left(k\right))^{2}kk^{\prime},$
23.7.3 $\wp\left(\tfrac{1}{2}\omega_{3}\right)=e_{3}-\sqrt{(e_{1}-e_{3})(e_{2}-e_{3})}% =e_{3}-\omega_{1}^{-2}(K\left(k\right))^{2}k,$
##### 5: 23.13 Zeros
###### §23.13 Zeros
For information on the zeros of $\wp\left(z\right)$ see Eichler and Zagier (1982).
##### 6: 23.10 Addition Theorems and Other Identities
###### §23.10(ii) Duplication Formulas
(23.10.8) continues to hold when $e_{1}$, $e_{2}$, $e_{3}$ are permuted cyclically. …
##### 7: 23.4 Graphics
###### §23.4(i) Real Variables
Line graphs of the Weierstrass functions $\wp\left(x\right)$, $\zeta\left(x\right)$, and $\sigma\left(x\right)$, illustrating the lemniscatic and equianharmonic cases. …
###### §23.4(ii) Complex Variables
Surfaces for the Weierstrass functions $\wp\left(z\right)$, $\zeta\left(z\right)$, and $\sigma\left(z\right)$. …
##### 8: 23.21 Physical Applications
###### §23.21 Physical Applications
The Weierstrass function $\wp$ plays a similar role for cubic potentials in canonical form $g_{3}+g_{2}x-4x^{3}$. …
###### §23.21(iii) Ellipsoidal Coordinates
where $x,y,z$ are the corresponding Cartesian coordinates and $e_{1}$, $e_{2}$, $e_{3}$ are constants. …
##### 9: 23.23 Tables
###### §23.23 Tables
2 in Abramowitz and Stegun (1964) gives values of $\wp\left(z\right)$, $\wp'\left(z\right)$, and $\zeta\left(z\right)$ to 7 or 8D in the rectangular and rhombic cases, normalized so that $\omega_{1}=1$ and $\omega_{3}=ia$ (rectangular case), or $\omega_{1}=1$ and $\omega_{3}=\tfrac{1}{2}+ia$ (rhombic case), for $a$ = 1. …05, and in the case of $\wp\left(z\right)$ 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).
##### 10: 23.9 Laurent and Other Power Series
###### §23.9 Laurent and Other Power Series
$c_{2}=\frac{1}{20}g_{2},$
For $j=1,2,3$, and with $e_{j}$ as in §23.3(i),
23.9.6 $\wp\left(\omega_{j}+t\right)=e_{j}+(3e_{j}^{2}-5c_{2})t^{2}+(10c_{2}e_{j}+21c_% {3})t^{4}+(7c_{2}e_{j}^{2}+21c_{3}e_{j}+5c_{2}^{2})t^{6}+O\left(t^{8}\right),$
Also, Abramowitz and Stegun (1964, (18.5.25)) supplies the first 22 terms in the reverted form of (23.9.2) as $1/\wp\left(z\right)\to 0$. …