# Weierstrass elliptic-function form

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##### 2: 1.13 Differential Equations
###### §1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms
This is the Sturm-Liouville form of a second order differential equation, where denotes $\frac{\mathrm{d}}{\mathrm{d}x}$. Assuming that $u(x)$ satisfies un-mixed boundary conditions of the form
##### 3: 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. … Another form is obtained by identifying $e_{1}$, $e_{2}$, $e_{3}$ as lattice roots (§23.3(i)), and setting …
##### 4: 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 $1/\wp\left(z\right)\to 0$. …
##### 6: 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. …
##### 7: 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 $\int{\wp}^{3}\left(z\right)\,\mathrm{d}z=\frac{1}{120}\wp'''\left(z\right)-% \frac{3}{20}g_{2}\zeta\left(z\right)+\frac{1}{10}g_{3}z.$
##### 8: 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 $\wp\left(\tfrac{1}{2}\omega_{3}\right)=e_{3}-\sqrt{(e_{1}-e_{3})(e_{2}-e_{3})}% =e_{3}-\omega_{1}^{-2}(K\left(k\right))^{2}k,$
##### 9: 23.13 Zeros
###### §23.13 Zeros
For information on the zeros of $\wp\left(z\right)$ see Eichler and Zagier (1982).